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REVOLVING  VECTORS 


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TORONTO 


REVOLVING  VECTOES 


WITH    SPECIAL    APPLICATION    TO 


ALTERNATING    CURRENT     PHENOMENA 


BY 


GEORGE  W.  PATTERSON,  S.B.  PH.D. 

PROFESSOR    OP   ELECTRICAL,    ENGINEERING,    UNIVERSITY    OF    MICHIGAN, 
MEMBER  AM.   INST.   EL.   ENG.,      MEMBER  AM.    PH.   SOC-,   ETC. 


f  mrk 
THE  MACMILLAN  COMPACT 

LONDON:  MACMILLAN  &  CO.,  LTD. 
1911 

All  rights  reserved 


COPYRIGHT,  1911, 
BY  THE  MACMILLAN  COMPANY 


Set  up  and  electrotyped  September,  1911 


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Engineering 
Library 


PREFATORY   NOTE 


THE  use  of  complex  quantities,  i.e.,  quantities  part  real 
and  part  imaginary,  in  the  theory  of  alternating  currents 
has  been  greatly  developed  by  Dr.  Charles  P.  Steinmetz 
in  his  work  on  "  Alternating  Current  Phenomena."  It 
would  be  difficult  to  determine  the  influence  which  earlier 
writers,  from  the  time  of  Caspar  Wessel  down  to  Steinmetz's 
day,  have  had  in  laying  the  foundation  on  which  Stein- 
metz has  built.  It  is,  however,  fair  to  say  that  the  great 
advance  in  the  use  of  vector  methods,  both  algebraic  and 
geometric,  due  to  Dr.  Steinmetz,  justifies  us  in  calling 
their  application  to  alternating  current  phenomena  Stein- 
metz's  Method. 

Earlier  writers  used  complex  quantities  to  represent 
vector  quantities  algebraically.  Dr.  Steinmetz  extended  the 
application  so  as  to  include  harmonic  quantities.  As 
many  writers  on  electrical  subjects  are  prone  to  confuse 
vector  and  harmonic  quantities,  the  author  thinks  it 
necessary  to  distinguish  these  two  uses  of  complex  quan- 
tities, and  for  that  purpose  he  starts  with  the  vector  use 
and  later  takes  up  the  harmonic  use.  In  addition,  sub- 
traction and  certain  cases  of  multiplication  and  division, 
correct  results  are  obtained  by  treating  harmonic  quantities 
as  vector  quantities;  but  in  other  cases  of  multiplication 
(such  as  multiplication  of  e.m.f.  and  current  to  obtain 
power)  and  division  (such  as  dividing  power  by  e.m.f. 
to  get  current)  incorrect  results  are  obtained  unless  arbi- 

iii 


346391 


iv  PREFATORY  NOTE 

trary  rules  of  multiplication  and  division  are  introduced. 
It  therefore  is  necessary  thoroughly  to  examine  the  funda- 
mentals of  these  uses  of  complex  quantities,  and  to 
deduce  the  laws  of  addition,  subtraction,  multiplication, 
and  division,  as  applicable  to  vector  quantities  and  to 
harmonic  quantities  whether  simple  (electromotive  force 
and  current)  or  compound  (power),  and  also  to  such  non- 
harmonic  quantities  as  resistance,  capacity,  inductance, 
etc.,  in  connection  with  harmonic  quantities. 

There  are  two  methods  employed  by  electricians  using 
the  complex  quantity  notation.  The  older  method  due  to 
Dr.  Steinmetz  is  expressed  in  graphical  form  by  the  wave 
diagram.  The  other  method  uses  the  so-called  crank 
diagram.  In  both  methods  counter-clockwise  rotations  are 
used,  though  the  formulae  have  led  some  persons  to  think 
that  Dr.  Steinmetz  has  used  clockwise  rotations.  It  is 
true  that  the  imaginary  terms  in  the  resulting  formulae 
have  opposite  signs.  In  reading  Dr.  Steinmetz's  works  no 
confusion  need  result  for  one  accustomed  to  the  crank 
diagram  method  if  the  differences  are  kept  in  mind.  It 
has  seemed  to  the  author  that  the  crank  diagram  method 
suits  his  purpose  better,  and  consequently  it  will  be  used 
in  this  book. 


TABLE  OF  CONTENTS 


PAGE  SECTIONS 

PEEFATORY   NOTE iii  , 

CHAPTER         I.  Rotary  power  of  roots  of  minus  one 1  1-11 

CHAPTER      II.  Rotary  power  of  imaginary  exponents.  .  .    12  12,  13 

CHAPTER     III.  Position  of  a  point  in  a  plane 14  14 

Uniform  circular  motion 15  15 

Effect  of  damping,  spiral  motion 18  16 

CHAPTER      IV.  Simple  harmonic  quantities 21  17 

Harmonic  electromotive  force 22  18,  19 

Harmonic  current,  impedance 24  20,  21 

Harmonic  electromotive  forces  in  series  .  .  30  22 

Problem  of  the  divided  circuit 32  23,  24 

Resolution  into  components  . 35  25,  26 

Use  of  a  symmetrical  pair  of  triangles ....   38  27 

CHAPTER        V.  Product  of  two  harmonic  quantities 39  2 8,  29 

Power  in  simple  circuits 40  30-33 

Power  in  circuits  of  more  than  one  phase .   45  34 

Balanced  two-phase  circuit 46  '   35 

Balanced  three-phase  circuit 46  36 

Balanced  four-phase  circuit 47  37 

Balanced  six-phase  circuit 48  38 

Balanced  polyphase  circuits  in  general.  .  .  48  39 

Unbalanced  polyphase  circuits 50  40 

CHAPTER      VI.  Non-harmonic  currents 52  41 

Oscillatory  discharge  of  a  condenser 52  42,  43 

Non-oscillatory  discharge  cf  a  condenser  59  44 
Phenomena  observed  on   closing  the  cir- 
cuit, starting  term 60  45-48 

General  remarks 67  49 

v 


vi  TABLE  OF  CONTENTS 

PAGE  SECTIONS 

CHAPTER    VII.  Compound     harmonic     current,    electro- 
motive force  and  power 68  50 

Use  of  Fourier  series 68  51,  52 

Power  factor 71  53,  54 

CHAPTER  VIII.  Interlinked  circuits,  mutual  inductance  .  .  74  55 
Ohm's  law  extended  to  mutually  induc- 
tive circuits 75  56-58 

Faraday's  ring 76  59 

Concerning  lines  of  force 78  60 

Ratio  of  transformation 79  61,  62 

Transformer  diagrams,  lagging  current. .  .   80  63 

Exciting  current,  core  losses 82  64 

Effect  of  flux  leakage 83  65 

Transformer  equations 83  66 

Transformer  diagrams,  leading  current ...   84  67 

Difficulty  found  in  exponential  expression .  85  68 

Conclusion 85  69 

Index..  .  87 


EEVOLVING  VECTORS 


REVOLVING  VECTORS 


CHAPTER  I 
ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE 

§  1.  Before  the  year  1797  algebraic  expressions  were 
used  to  represent  magnitudes  only,  but  in  that  year 
a  Danish  surveyor,  Caspar  Wessel,  presented  a  memoir  to 
the  Royal  Academy  of  Sciences  and  Letters  of  Denmark, 
entitled  "On  the  Analytical  Representation  of  Direction1." 
Wessel's  memoir  laid  the  foundation  for  vector  anaylsis, 
for  the  theory  of  functions  of  a  complex  variable,  and  for 
Steinmetz's  method  for  alternating  current  phenomena. 
In  this  paper  Wessel  introduced  \/—  1  as  the  sign  of  per- 
pendicularity, and  used  the  letter  £  to  indicate  this  use 
of  the  imaginary  unit.  It  is  now  common  to  use  the  letters 
i  or  j  for  V7— 1.  He  showed  how  a  quantity  might  be 
represented  both  in  magnitude  and  direction  by  an  algebraic 
expression,  and  made  it  clear  that  he  used  the  sign  of  per- 
pendicularity not  as  a  factor  in  the  strict  algebraic  sense, 
but  rather  as  an  operator  functioning  to  rotate  the  mag- 

1  Om  Direktionens  analytiske  Betegning,  et  Forsog  anvendt  fornem- 
melig  til  plane  og  sphaeriske  Polygoners  Oplosning;  read  March  10, 
1797;  Memoirs  of  the  Academy,  Vol.  V,  1799;  republished  in  French 
by  the  Academy  1897;  see  also  Beman,  Proc.  A.  A.  A.  S.,  1897,  Vol. 
XLVI,  p.  33. 


2  REVOLVING  VECTORS 

nitude  for  which  the  rest  of  the  algebraic  quantity  stood, 
through  an  angle  of  90°. 

Until  the  discovery  by  Wessel  of  the  use  of  V  —  1  as 
the  sign  of  perpendicularity,  this  symbol  when  occurring 
in  a  problem  had  always  been  taken  as  a  sign  indicating 
the  impossibility  of  the  problem,  just  as  we  are  accustomed 
to  view  the  answer  obtained  as  absurd  if,  for  example,  we 
are  required  to  divide  seven  apples  into  piles  having  ten 
apples  in  each.  In  early  ages  minus  quantities  were  un- 
known, and  solutions  involving  the  indication  of  a  minus 
quantity  were  taken  to  indicate  an  impossibility.  We 
now  think  it  fair  to  hold  our  minds  open  to  some  new 
meaning  to  be  assigned  to  symbols  which  formerly  were 
meaningless  if  not  absurd. 

Wessel's  memoir  met  the  same  fate  as  many  others 
written  in  advance  of  their  time,  such,  as  Green's  essay  in 
which  the  potential  function  was  christened,  and  Gibbs's 
essays  in  which  the  foundation  of  the  thermodynamics  of 
the  voltaic  cell  was  laid;  for  Wessel's  paper  was  put  to 
sleep  in  the  printed  memoirs  of  the  Academy,  its  slumber 
not  to  be  disturbed  until  long  years  after,  when  Wessel's 
ideas  had  been  rediscovered  by  other  men  such  as  Argand, 
Gauss,  Cauchy,  Frangais,  and  Gergonne. 

§  2.  To  show  how  little  prepared  the  mathematical 
world  was  for  Wessel's  use  of  V—  1,  it  is  interesting  to 
find  that  Cauchy  l  as  late  as  1844  said: 

"Every  imaginary  equation  is  naught  else  than  the 
symbolic  representation  of  two  equations  between  real 
quantities.  The  employment  of  imaginary  expressions 
by  permitting  us  to  replace  two  equations  by  a  single 
one,  often  offers  the  means  of  simplifying  calculations 
and  of  writing  in  abridged  form  quite  complicated  re- 
sults. Such  indeed  is  the  principal  motive  for  con- 

1  Cauchy,  Exercise  d'analyse  et  de  physique  mathematique,  Tome 
III,  p.  361. 


ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE     3 

tinuing  our  use  of  these  expressions,  which  taken  literally 
and  interpreted  according  to  generally  established  con- 
ventions signify  nothing  and  are  without  sense." 

Professor  Durege  of  Prague  says  in  the  introduction  to 
his  book  on  "  The  Theory  of  Functions  of  a  Complex  Vari- 
able"1: 

"  The  work  of  deMoivre,  Bernoulli,  the  two  Fagnanos, 
d'Alembert,  Euler,  and  others  was,  on  the  whole,  looked 
upon  more  as  scientific  foolery  (Spielereien  fur  blose 
Curiosa),  and  that  it  was  entitled  to  appreciation  of 
worth  only  in  proportion  as  it  lent  useful  means  to 
help  in  other  investigations." 

§  3.  To  prepare  for  a  meaning  to  an  even  root  of  a 
negative  quantity,  it  may  be  useful  to  consider  how  a 
negative  quantity  was  transferred  from  the  absurd  and 
impossible  to  the  category  of  real  and  possible  quantities. 
We  are  all  in  agreement  that  no  quantity,  in  the  strict 
sense,  can  be  a  quantity  at  all  if  its  magnitude  is  less  than 
zero.  How  are  we  then  to  understand  the  negative  sign, 
if  a  negative  quantity  is  to  have  real  meaning?  The 
solution  of  the  puzzle  is  illustrated  by  means  of  such  a 
problem  as  this:  The  point  A  is  five  miles  east  of  B}  the 
point  C  is  ten  miles  east  of  B.  How  many  miles  is  A  east 
of  C?  The  result  is  — 5  miles.  The  old  interpretation  of 
the  result  is  that  the  answer  is  absurd,  for  A  is  not  east 
of  C  at  all.  The  modern  intepretation  is  that  the  answer 
is  not  absurd  and  that  the  —5  miles  is  to  be  understood 
as  5  miles  west.  Or  put  in  another  way,  the  negative  sign 
is  not  to  be  understood  as  compelling  us  to  consider  a 
distance  less  than  nothing,  but  simply  that  the  minus 
sign  is  an  operator  which  functions  to  change  our  east- 
ward sense  of  counting  into  the  opposite  or  westward. 

1  Dur&ge,  "  Theorie  der  Fuoktionen  einer  complexen  verander- 
lichen  Grosse,"  p.  2. 


4  REVOLVING  VECTORS 

We  may  then  consider  that  what  has  been  taken  as  multi- 
plication by  —1  is  not  really  multiplication  at  all,  but 
merely  the  use  of  an  operator  turning  an  eastward  through 
180°  into  a  westward  sense  of  counting. 

§  4.  In  an  analogous  way,  let  us  consider  what  would 
happen  if  an  operator  could  be  found  which  on  being 
applied  to  eastward  sense  would  change  it  to  northward 
and  on  being  applied  a  second  time  would  change  north- 
ward to  westward,  and  so  on  with  successive  applications 
of  this  operator,  changing  westward  to  southward,  and  south- 
ward to  eastward.  That  V— 1  is  such  an  operator  was 
discovered  by  Wessel,  and  as  before  mentioned,  he  called 
it  the  sign  of  perpendicularity,  for  he  found  that  V—  1 
used  twice  would  cause  a  reversal  of  direction  or  a  rotation 
of  180°;  and  what  was  more  natural  than  to  assume  that 
one  application  would  produce  a  rotation  of  90°  ?  To 
avoid  ambiguity  between  the  two  senses  in  a  plane  through 
which  the  180°  rotation  to  produce  a  reversal  might  be 
taken,  we  may  agree  to  consider  the  rotation  to  take  place 
counter-clockwise.  We  thus  have 

\^1  East  =  North, 
\/^lV^l  East  =  \/^l  North  =  West, 

=  V^T  West  =  South, 
=  V^l\^:n\/^l  North 

^l  South  =  East. 

To  abridge  the  notation,  but  not  compromising  our 
attitude,  we  may  write; 


ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE     5 

§  5.  Although  no  contradictory  complication  has  resulted 
from  the  use  of  V^~I  as  a  90°  rotator,  the  reader  may  have 
grave  doubts  of  the  safety  of  using  it  in  all  cases  as  WessePs 
sign  of  perpendicularity.  His  confidence  may  be  increased 
by  showing  that  analogous  assumptions  of  rotary  powers, 
given  to  other  roots  of  —  1,  are  free  from  contradictions. 
To  this  end  let  us  examine  %/  —  1,  which  we  should  expect 
to  be  endowed  with  the  ability  to  rotate  through  60° , 
as  three  applications,  as  an  algebraic  multiplier,  would  be 
equivalent  to  multiplying  by  —  1.  How  about  it  as  an 
operator  f  Let  us  assume  that 


or 


This  equation  has  three  roots,  as  follows 


Assuming  that  3/—1  may  be  used  as  an  operator  to 
an  eastward  direction  as  before,  we  have  three  results  : 

East  =  (i  +  i  \/3  \/^T)  East  =  i  East  +  i\/3"  North, 


or 


or 


~^l  East  =  -  1  East  =  West, 
^l  East  =  ft  -  i\/3  \/^l)  East  =  J  East  +  ^Vs  South. 


From  Fig.  1  it  is  evident  that  the  first  result  produces 
a  rotation  of  60°  without  change  in  magnitude,  for  the 
sine  of  60°  is  JV?  and  the  cosine  is  J.  The  second  result 
is  a  rotation  of  180°,  simply  changing  eastward  into  west- 
ward. The  third  result  is  either  a  backward  rotation  of 


6  REVOLVING  VECTORS 

60°  or  a  forward  (counter-clockwise)  rotation  of  300°. 
All  on  being  repeated  for  the  third  time  produce  reversals 
of  direction;  for  they  give  one-half  a  rotation,  one  and 
one-half  rotations,  and  two  and  one-half  rotations  respect- 
ively, all  being  taken  as  counter-clockwise. 

To  avoid  confusion  we  shall  take  3/  —  I  to  be  an  operator 
endowed  with  the  power  of  producing  a  rotation  of  60°. 


East 


FIG.  1. 


§  6.  We   may  proceed  in   a   similar  way  to  show  that 
V— 1  may  be  used  to  produce  a  rotation  of  45°.     Let 


or 


This  equation  has  four  solutions,  as  follows 


x=  - 


ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE     7 

Applying  them  successively  to  an  eastward  direction 
as  before  we  have  the  four  results, 

V^  East  =  iv/2  East+i\/2  North  =  Northeast, 
or  =  i\/2  West  +  J\/2  North  =  Northwest, 

or  '  =  J V2  West  +  £\/2  South = Southwest , 

or  =  i\/2~  East + i\/2"  South  =  Southeast . 

They  are  equivalent  to  rotations  of  45°,  135°,  225°  or 
315°,  and  all  on  being  repeated  for  the  fourth  time  pro- 
duce reversals  (without  or  with  extra  complete  rotations) . 

§  7.  In  the  same  way  all  other  roots  of  —  1  may  be 
examined  (for  all  are  known),  and  in  every  case  these 
roots  will  be  found  to  be  endowed  with,  the  property  that 
as  operators  they  will  produce  rotations  which,  on  being 
repeated  to  the  number  of  times  indicated  by  the  index 
of  the  root,  will  produce  reversals.  In  general  for  the 
root  giving  smallest  rotation 

\/—  1  =  cos  —  +  V—  1  sin  — . 
n  n 

Indicating  V  —  1  by  /,  we  may  write  this  equation 

-  7C 

?  n  =  cos— +  ?  sin— . 
n  n 

There  are  also  other  roots  giving  larger  rotations.  The 
general  expression  for  all  roots  is  as  follows: 

,  .   .     (2m+l)7r 


where  ra  stands  for  zero  or  any  positive  or  negative  whole 
number.  It  will  be  shown  below  that  n  need  not  be  a 
positive  whole  number.  In  fact  it  may  be  positive  or 
negative,  a  whole  number,  or  a  proper  or  improper  fraction, 


8  REVOLVING  VECTORS 

or  indeed  any  number  whatever,  and  we  shall  even  make 
use  of  it  as  a  variable  quantity  in  alternating  current 
applications. 

§  8.  Let   us   consider   two   operators,    able  to  produce 
rotation  through  angles  indicated  by  6  and  0, 

A  =  cos  0+/sin  6, 
5  =  cos  0+/sin  $. 

Multiplying  them  together,  we  get  an  operator  A  -  B, 
A.  B  =  cos  6  cos  0—  sin  6  sin  0  +  j(sin  6  cos  0  +  cos  6  sin  0) 
=  cos  (0+0)+/sin 


This  new  operator  has  the  power  of  producing  a  rota- 
tion through  the  sum  of  the  angles  0+0.  It  is  to  be 
remembered  that  /2=  —  1. 

If  we  divide  one  by  the  other,  we  obtain 

A     cos  0  +    sin  6 


_ 
B     cos  0  +  j  sin  (/> 

cos  6  cos  0  +  sin  0  sin  0  +  /(sin  0  cos  0—  cos  0  sin  0) 
cos2  0+sin2  (f> 

=  cos(0—  0)  +  /sin(0—  0). 

A 
From  which  it  appears  that  -=r-  is  an  operator  producing 

a  rotation  through  the  angle  0  —  0. 

In  a  similar  way  to  the  multiplication  above,  if  0  and 
0  are  equal,  we  have 

A2  =  cos  20  +/  sin  20, 

^.3  =  cos30+jsin30- 
or  in  general, 

An  =  cos  n6+j  sinn0=(cos  0+/sin0)n. 


ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE  9 

If  n6=  equals  180°  or  TC,  we  have 


n  =     o  =  cos 
and 


From  this  follows  the  value  of  \/—  I  given  above,  viz., 

n/  -  -  7T        .     .      7T 

V  —  1  =  cos  —  h  3  sin  —  . 
n  n 

The  general  value  given  above  may  be  reached  in  an 
analogous  manner,  for  if  m  is  any  whole  number  (positive 
or  negative)  or  zero,  we  have 


cos 
and 

n,  -  - 

V  —  1  =  cos  -  -  -  —  \-  i  sin 


This  last  expression,  though  it  appears  to  have  an 
indefinitely  great  number  of  different  values,  in  fact  has 
only  n  different  values,  if  n  is  a  whole  number;  for  the 
values  of  the  cosine  and  sine  repeat  after  n  different  values, 
it  being  evident  that  if  m  =  n, 


. 

cos  --  \-i  sin  -  =  cos  —  \-i  sin—. 
n  n  n  n 


If 

(2 

cos  -  -  •£-  +  1  sin 
«* 

and  so  on. 


n 


cos  -  -  -  —  -  ?  sin  -  -  •  —  —, 
n  n 


§  9.  If  n  is  a  fraction  equal  to  —  ,  which  may  be  a  proper 


10  REVOLVING  VECTORS 

or   improper   fraction    and   positive    or   negative,   both   p 
.and  q  being  whole  numbers,  we  shall  have 

P. 

i/ — 7      .—  an      .   .    an 

V-l  =  ?  P  =  cos  —  +  ?  sm  — : 
P  P 

for  on  raising  this  expression  to  the  power  p,  we  obtain 
j  29  =  cos  qn + j  sin  qx. 

If  q  is  an  odd  number  j2q=cos  qn  =  —  1,  and  if  q  is  an 
even  number  j2q  =  cos  qn  =  +1.  In  both  cases  sin  qx  is 
zero.  If  q  is  negative  the  same  result  follows. 

§  10.  If  n  is  a  number  which  is  neither  whole  nor  a 
proper  or  improper  fraction,  we  may  by  the  doctrine  of 
limits  have  confidence  in  assuming  that  V— 1  will  have 
a  value  between  V— 1  and  \/  —  1,  where  r<n<s,  and 
r  and  s  are  whole  numbers  or  fractions  very  close  to  one 
another  in  value. 

As  we  may  always  find  whole  numbers  or  fractions, 
one  larger  and  one  smaller  and  different  from  n  by  amounts 
less  than  any  assigned  amount,  in  the  limit  we  may  find 
the  value  of  \/—  1  with  as  high  a  degree  of  precision  as 
desired.  We  may  therefore  have  confidence  that  n  may 
be  a  continuously  varying  quantity,  say  a  function  of  the 
time.  For  example  let  to  be  an  angular  velocity  and  t  be 
time.  We  may  write  the  following  equation : 

TC_  2<at 

0>t; ,~ 

V  — l  =  j     =cos  cot  +  1  sm  (ot. 

This  equation  expresses  a  variable  operator  which 
functions  to  rotate  any  vector  to  which  it  is  applied  with 
a  counter-clockwise  angular  velocity  co.  In  the  case  of 
clockwise  rotation,  substituting  —  co  for  co.  we  obtain 

JT  _2(0t 

bit/ ,  K  .       . 

V  —  l  =  =cos  cot  —    sin  cot. 


ROTARY  POWER  OF  ROOTS  OF  MINUS  ONE         11 

It  should  be  had  in  mind  that  reversing  the  sign  of 
an  angle  does  not  affect  the  cosine,  but  does  reverse  the 
sine. 

§  11.  The  formulae  for  rotating  operators,  as  will  be 
shown  in  the  next  chapter,  may  be  more  conveniently 
expressed  as  powers  of  the  base  e  of  the  Napierian  system 
of  logarithms,  as  follows  : 


e"=  cos  (o+   sn  a)j 
and 

fi-jorf_cos  ut—    sin  cut. 


CHAPTER  II 
ROTARY  POWER  OF  IMAGINARY  EXPONENTS 

§  12.  It  has  been  shown  in  the  previous  chapter  that 
powers  or  roots  of  the  imaginary  unit,  V  —  1,  or  /,  may 
be  used  to  obtain  an  operator  which  can  function  to  rotate 
a  vector  quantity  either  through  a  stated  angle  or  through 
an  angle  increasing  continuously  with  the  time.  These 
operators  may  be  more  conveniently  expressed  as  imaginary 
powers  of  e,  the  base  of  the  Napierian  system  of  logarithms. 
The  mathematical  formulae  involved  were  already  old  and 
well  known  in  Wessel's  day  and  are  to  be  found  in  Euler's 
memoirs. 

If  we  expand  e  x,  sin  z  and  cos  x  in  powers  of  x  we 
obtain  : 


rr2 

_ 


It  is  evident  that  if  fa  is  substituted  for  x  we  shall  have 


cos  x  +  j  sin  x. 

12 


ROTARY  POWER  OF  IMAGINARY  EXPONENTS        13 

Referring  to  Chapter  I  it  is  evident  that  the  rotating 
operator  may  take  any  one  of  four  equal  forms  : 


20 


in  case  rotation  is  to  take  place  through  a  definite  angle 
6]  or  in  case  the  rotation  is  to  be  continuous  and  a  function 
of  the  time  the  four  equal  forms  may  be  written: 


=£/«*  =  cos  ajt+jsin.  cut, 

in  which  t  is  the  time  and  to  the  angular  velocity. 

§  13.  If  an  operator  consisting  of  the  sum  of  two  operators 
which  used  singly  would  produce  rotations  equal  in  mag- 
nitude but  opposite  in  sense,  is  used  on  a  vector,  the  oper- 
ator reduces  to  a  simple  factor  causing  the  vector  to  follow 
the  law  of  simple  harmonic  motion.  The  expression  for 
such  an  operator  is: 


a  result  which  might  have  been  obtained  directly  from 
Euler's  formula  for  the  cosine, 


cos  6  = 


CHAPTER  III 
POSITION  OF  A  POINT  IN  A  PLANE 

§  14.  Many  applications  of  complex  quantities  with  a 
vector  meaning  might  be  made.  It  is  believed,  however, 
that  the  following  will  suffice  to  illustrate  their  use. 

The  position  of  a  point  in  a  plane  may  be  determined 
by  rectangular  coordinates.  Using  as  before  /  for  WessePs 
sign  of  perpendicularity,  the  position  of  the  point  P  with 
reference  to  the  point  0  taken  as  the  origin  of  coordinates 
as  shown  in  Fig.  2,  has  the  following  expression: 

P-a+jb, 


a 
FIG.  2. 


Q 


a  being  the  horizontal  and  b  the  vertical  projection  of  the 
line  of  length  p,  connecting  0  and  P.  We  have  by  geometry 
that  lo2  =  a2+62,  a=pcos6,  and  b=psmd.  Using  polar 
coordinates,  the  equation  becomes 


p  cos  6+jp  sin  6=  (cos  d+j  sin  6) p. 


14 


UNIFORM   CIRCULAR  MOTION  15 

Using  the  notation  of  the  last  chapter,  we  have 
P  =  e^p  =  (cos  0+j  sin  0)p. 

Wessel's  idea  of  the  analytical  representation  of  direc- 
tion has  an  illustration  in  each  of  the  equal  operators 
£*e  and  cos#+/sin#.  Each  has  a  magnitude  unity,  and 
each  may  be  considered  to  have  the  sole  effect  of  specifying 
a  direction  differing  by  an  angle  6  from  the  direction  (hori- 
zontal) taken  as  standard.  It  is  permissible  also,  as  pre- 
viously done,  to  consider  e'°  and  cos  0+/sin  6  as  operators 
functioning  to  turn  p  from  a  horizontal  position  to  that 
indicated  in  the  figure.  The  whole  expression  for  P, 


has  both  magnitude  and  direction  expressed,  the  first  form, 
a+jb,  expressing  by  a  both  magnitude  and  horizontal 
direction  and  by  b  both  magnitude  and  horizontal  direction, 
the  latter  being  rotated  into  a  vertical  direction  by  the 
operation  of  /.  In  the  later  expressions  the  operators,  or 
analytical  expressions  of  direction,  s1'6  and  cos0-f/sin0, 
are  expressed  separately  from  the  magnitude  p  which,  if 
standing  alone,  would  have  been  understood  as  a  horizontal 
magnitude. 

UNIFORM   CIRCULAR   MOTION 

§  15.  Another  simple  illustration  may  be  taken  from 
uniform  circular  motion,  one  of  the  simplest  motions  met 
with  in  physics  and  which  we  shall  use  later  in  connection 
with  harmonic  quantities. 

First  let  us  consider  the  position  of  a  point  moving 
about  the  circumference  of  a  circle.  Let  the  radius  of  the 
circle  be  R  and  the  angular  position  of  the  radius  be  expressed 
in  terms  of  angular  velocity  a>  and  the  time  t  elapsed  since 


16  REVOLVING  VECTORS 

the  radius  was  horizontal  and  directed  to  the  right  (Fig.  3). 
Let  us  take  the  center  of  the  circle  0  as  origin  of  coordi- 
nates. We  have  then 


P  =  £iutft  =  (cos 

If  for  any  reason  it  is  desirable  to  take  as  origin  any 
other  point  with  coordinates  a  and  jb,  and  to  measure 
angular  position  from  a  radius  making  an  angle  0  with  the 
horizontal,  we  would  have 

>t-0)+j8m  (wt-ff))R-a-jb. 


FIG.  3. 

The  introduction  of  an  eccentric  origin  (a,  jb)  and  an 
epoch  (0)  introduces  no  real  difficulty,  though  it  com- 
plicates the  expression. 

Consider  now  the  velocity  V  of  the  point  P.  Evidently 
we  have  (using  the  earlier  expression  for  P) , 


UNIFORM  CIRCULAR  MOTION  17 

This  shows  that  the  magnitude  of  the  velocity  is  wR, 
and  the  phase  90°  or  —  ahead  of  the  phase  of  P,  both  well- 

known  facts  of  uniform  circular  motion.  As  the  last  trans- 
formation may  have  difficulties  for  some  readers,  it  is 
well  to  note  that,  as  given  earlier, 


and  if  cut  equals  —  ,  this  equation  reduces  to 
2 


therefore 


It  is  also  of  advantage  in  differentiating  cos  cot +j  sin.  tot 
not  to  change  from  cosine  to  sine  and  vice  versa,  but 

rather  to  advance  ths  phase  by  — ,  which  comes  to  the 

2i 

same  thing.  Changing  from  cosine  to  sine  and  vice  versa 
in  differentiating  harmonic  quantities  conceals  the  change 
of  phase  from  immediate  notice,  and  a  clear  understanding 
of  phase  relations  is  desirable. 

If  the  origin  of  coordinates  is  not  at  the  center  of  the 
circle  and  if  there  is  an  epoch  angle  in  the  expression,  the 
second  expression  for  P, 

P  =  £^-^-a-/6=(cos  (ojt-6)+jsm(ojt-0))R-a-jb 
leads  to  a  value  of  V, 


18  REVOLVING  VECTORS 

The  acceleration  A  in  uniform  circular  motion  is  as 
follows  : 


n)+JBW  (a>t+n))a?R, 
for  the  simple  case,  and 


for  the  more  involved  case.  The  meaning  of  the  formulae 
is  that  the  magnitude  of  the  acceleration  is  a>2R;  and  its 
phase  tot+n  (or  wt+n—  6)  shows,  by  the  added  n,  that 
the  acceleration  is  directed  toward  the  center. 


EFFECT  OF  DAMPING,  SPIRAL  MOTION 

§  16.  Another  interesting  example  is  found  in  the  expo- 
nential  spiral  which  a  pendulum  started  in  motion  in  a 
horizontal  circular  path  will  follow  if  its  motion  is  damped 
in  proportion  to  its  velocity  (Fig.  4).  The  equation  for 
the  position  of  the  pendulum  is  as  follows,  if  we  take  the 
center  as  origin  and  assume  the  epoch  as  zero: 

In  this  expression  #£""'  is  the  magnitude  of  the  distance 
from  the  origin,  and  ^wi,  or  cos  tot +j  sin  cot,  is  the  ana- 
lytical expression  for  the  direction  of  the  line  from  0  to 
P.  Differentiating  P  with  respect  to  t}  we  get  the  velocity 
of  the  pendulum, 


—  (cos  tot + /  sin  tot) a\Re~at=  (jco  —  a)  e***-  atR. 

T.  /«•/  #  j  /  & 

.Let  tan0  =  — ,     sin<z>  = — .  -     ana     cosa>  =  — .  :. 

Va2  +  to2 


EFFECT  OF  DAMPING,  SPIRAL   MOTION  19 

The  expression  for  V  then  may  be  rewritten 


FIG.  4. 
The  acceleration  A  becomes 


These  equations  show  that  the  phase  of  the  velocity 
of  a  damped  circular  pendular  motion  is  —  -\-(f>  in  advance 


20  REVOLVING  VECTORS 

of  the  phase  of  the  position  of  the  pendulum,  i.e.,  $  more 
than  a  quadrant:  and  the  phase  of  the  acceleration  is  in 
advance  of  that  of  the  velocity  an  equal  amount.  The 
acceleration  is  not  directed  toward  the  center,  as  is  the 
case  in  uniform  circular  motion,  but  is  in  advance  of  a 
line  drawn  to  the  center  by  an  angle  2^>. 

The  real  part  of  the  above  expressions  is  applicable  to 
simple  pendular  motion  (in  a  vertical  plane)  or  to  the 
movement  of  a  ballistic  galvanometer  with  damping  of 
moderate  magnitude,  and  analogous  expressions  apply  to 
the  charge  and  current  in  the  case  of  the  oscillatory  dis- 
charge of  a  condenser  in  an  inductive  circuit. 

The  origin  need  not  be  taken  at  the  center  of  the  spiral, 
and  there  may  be  an  epoch  angle  if  for  any  reason  it  is 
considered  desirable  not  to  take  the  coordinates  as  assumed 
above.  The  complications  resulting  are  not  troublesome. 


CHAPTER  IV 
SIMPLE   HARMONIC   QUANTITIES 

§  17.  In  the  previous  chapters  we  have  used  complex 
quantities  in  connection  with  real  vectors  only.  In  this 
chapter  we  shall  make  use  of  vector  expressions  to  repre- 
sent simple  harmonic  quantities. 

In  Chapter  II  the  connection  was  shown  between  a 
simple  harmonic  motion  and  a  pair  of  circular  motions 
equal  in  magnitude  but  with  oppositely  directed  angular 
velocities.  Algebraically  this  connection  is  expressed  by 
Euler's  formula  for  the  cosine, 


cos  cut= 


As  is  well  known,  the  real  parts  of  £J'ut  and  e~'w  are 
identical  and  the  imaginary  parts  equal  and  opposite. 
We  therefore  have  the  relation 

^ 

£J<at    I    g  —  jtat 

cos  a>t=  —  —  -  -  =real  part  [V^] 


It  appears  from  this  expression  that  instead  of  expressing 
the  simple  harmonic  motion  as  the  sum  of  two  oppositely 
directed  uniform  circular  motions  which  are  equal  in  mag- 
nitude, we  might  equally  well  have  considered  the  simple 
harmonic  motion  as  the  real  part  of  a  single  uniform 
circular  motion  of  twice  the  magnitude  of  one  of  the  pair 
and  revolving  either  clockwise  or  counter-clockwise  as  one 
may  prefer.  This  statement  amounts  to  saying  that  a 
simple  harmonic  motion  is  the  projection  of  a  uniform 

21 


22 


REVOLVING  VECTORS 


circular  motion  on  the  diameter  of  the  circle,  or  as  many 
writers  say:  simple  harmonic  motion  is  the  apparent 
motion  of  a  point  in  uniform  circular  motion  when  viewed 
from  a  distant  point  in  the  plane  of  the  motion. 


HARMONIC  ELECTROMOTIVE  FORCE 

§  18.  Let  us  consider  an  electromotive  force  of  the  form 

e=E  cos  cot, 

which  may  be  considered  as  the  real  part  of  the  expression, 
E  =  £J^E=  (cos  cot  +j  sin  cot)E. 

This  equation  is  represented  graphically  by  OP,  the  radius 
of  the  circle  understood  to  be  revolving  counter-clockwise 
in  the  figure  (Fig.  5) . 

P 


o 


A  dot  over  or  under  a  symbol  will  be  understood  to 
mean  that  the  quantity  is  analogous  to  a  uniform  circular 
motion,  but  no  information  is  given  with  respect  to  the 
period  or  phase  of  the  variable.  It  must  always  be  had 
in  mind  that  only  the  real  part  of  the  complex  expression 
is  to  be  considered  seriously.  The  other  part  is  to  be 
looked  upon  as  scaffolding  about  a  building  in  process  of 
erection,  or  the  sawdust  in  a  box  of  torpedoes,  which  need 


HARMONIC  ELECTROMOTIVE   FORCE  23 

not  be  confused  with  the  building  or  the  torpedoes  them- 
selves, respectively,  for  the  imaginary  symbol  is  a  warning 
that  the  associated  term  is  to  be  disregarded.  Terms  in 
which  /  occurs  as  an  index  must  be  resolved  into  their 
real  and  imaginary  parts  before  the  latter  may  be  dis- 
regarded. 

No  error  will  be  made  in  adding  or  subtracting  such 
expressions,  for  the  real  part  of  the  sum  or  the  difference 
of  two  complex  quantities  is  the  sum  or  the  difference  of 
the  real  parts  only.  Multiplication  or  division  by,  or 
differentiation  with  respect  to,  any  real  quantity  cannot 
cause  any  confusion;  for  none  of  these  processes  can  change 
a  term  from  real  to  imaginary  or  vice  versa.  But  multi- 
plication, or  division  by,  or  differentiation  with  respect  to, 
any  imaginary  or  complex  quantity  is  apt  to  result  in 
confusion  unless  quite  arbitrary  rules  are  used  for  these 
operations.  As  a  rule  in  the  multiplication  of  two  simple 
harmonic  quantities,  we  may  not  use  the  whole  expression, 
but  only  the  real  parts.  As  an  example,  in  obtaining  the 
expression  for  power  by  multiplying  current  and  e.m.f., 
we  must  use  the  real  parts  only.  Power,  as  a  rule,  is  not 
a  simple  harmonic  quantity,  but  is  a  sum  (or  difference) 
of  a  constant  and  a  simple  harmonic  quantity  of  double 
frequency.  Dr.  Steinmetz  by  using  an  arbitrary  rule  for 
such  multiplication  obtains  the  average  value  of  the  power. 
As  it  may  be  shown  that  Dr.  Steinmetz's  rule  for  obtaining 
average  power  always  leads  to  the  right  result,  his  rule 
may  be  used  fearlessly  in  such  cases. 

§  19.  In  representing  harmonic  current  or  e.m.f.  by 
means  of  the  analytical  expression  for  a  revolving  vector, 
it  was  assumed  above  that  the  projection,  represented  by 
the  real  part  of  the  expression,  should  be  the  graphical 
representation  of  the  current  or  e.m.f.  respectively,  and 
therefore  diagrams  should  be  drawn  to  the  proper  scale. 
For  many  purposes  it  will  be  found  more  convenient  to 
change  the  scale  in  such  a  way  that  the  length  of  the  revolving 


24  REVOLVING  VECTORS 

vector  shall  represent  the  effective  (square  root  of  mean 
square)  value  which  is  indicated  by  an  ammeter  or  volt- 
meter in  the  respective  cases.  This  value  for  simple  har- 
monic cases  is  i\/2,  about  0.707,  times  the  maximum 
value.  The  analogous  equation  is  as  follows: 

E  =  ^wtV2E  =  (cos  tot+j  sin  cut)  \/2E, 

if  the  value  at  any  time  t  is  to  be  given  by  the  projection. 
As  it  is  only  rarely  that  we  desire  to  know  instantaneous 
values,  it  is  more  usual  to  use  the  former  expression 

E  =  Efa'E  =  (cos  tot+j  sin  ojt)E, 

and  understand  by  E  the  reading  of  the  voltmeter,  and 
in  case  instantaneous  values  are  ever  needed,  to  find  them 
by  multiplying  the  real  part  of  E  at  any  instant  by  V2. 
The  beginner  must  early  master  the  difficulties,  introduced 
or  avoided  by  using  or  suppressing  V2  in  the  formulae, 
and  be  on  his  guard  to  avoid  misunderstanding  various 
writers.  It  may  be  said  that,  as  a  rule,  harmonic  currents 
or  electromotive  forces  are  expressed  in  effective  values, 
while  harmonic  magnetic  fields  are  expressed  in  maximum 
values.  Power  is  as  a  rule  expressed  in  average  values. 
The  reasons  for  these  apparently  arbitrary  choices  the  more 
advanced  student  has  probably  already  learned.  We 
cannot  take  space  here  to  go  into  the  matter  further, 
and  must  be  content  with  the  bare  statement. 

HARMONIC  CURRENT,  IMPEDANCE 

§  20.  If  the  current,  as  well  as  the  electromotive  force, 
follows  an  harmonic  law,  and  lags  behind  the  e.m.f.  by 
a  phase  difference  represented  by  the  angle  0,  we  may 
write 

i  =  I  cos  (tot— 6), 


HARMONIC  CURRENT,   IMPEDANCE  25 

where  7  is  the  maximum  value  of  i.  If  the  circuit  has  a 
resistance  R  and  an  inductance  L,  and  is  not  complicated 
by  capacity  or  mutual  inductance,  and  includes  no  motors 
or  sources  of  e.m.f.,  Ohm's  law  modified  for  varying  currents 
gives 

D-          Tdi         V 

e  =  Ri-\-L-r  =  E  cos  cot. 
dt 

Substituting  the  value  for  i,  as  given  above,  in  the  last 
equation,  we  obtain 

E  cos  cot  =  RI  cos  (cot  —  0)  +  Lcol  cos  I  cot  —  0  -H  —  ) . 

If  this  equation  is  true  for  any  and  all  times,  it  is  evi- 
dent that 

(\ 
cot  —  6+—}. 

From  this  it  follows  that  the  next  equation  is  true, 
E  =  (cos  cot  +  /  sin  cot)  E  =  RI  (cos  (cot  —  0)  +  /  sin  (cot  -  0) ) 


and  remembering  that  j2=  —  l,  we  have  by  simple  trans- 
formations 


E=(R+jLco)(cos  (cot-6)  +/  sin  (^-#))/  =  £>w#, 


or 


Substituting  the  exponential  for  the  cosine  and  sine  expres- 
sion, we  have 


The  diagram  (Fig.  6)  shows  the  relations  analytically 
expressed  by  the  equations.  The  projection  of  E  on  the 
horizontal  axis  (axis  of  real  values)  equals  the  sum  of  the 


26 


REVOLVING  VECTORS 


projections  of  RI   and  jLa>I,  as  expressed  by  the  earlier 
equation, 

e  =  E  cos  wt  =  RI  cos  (a)t  —6)+  La>I  cos  f  cot  —  0  +  —  )  . 

As  Lwl  cos  (  a)t—  #4-^-)  is  a  quarter    period  in  advance  of 
RI  cos  (tut—  6),  it  is  evident  that  the  triangle  is  a  right 


triangle.     By  geometry  we  then  have  for  the  magnitudes 
involved, 


and 


E 


This  ratio  between  E  and  /  is  called  the  impedance  of  the 
circuit  and,  in  the  extension  of  Ohm's  law  to  alternating 
currents,  plays  the  part  that  resistance  does  for  direct 
currents.  Impedance  is  measured  in  ohms,  just  as  though 
it  were  a  real  resistance. 

In  a  similar  way  from  the  uniform  circular  formula, 
we  have 


)'9  f 


HARMONIC  CURRENT,  IMPEDANCE  27 

The  complex  constant  R+jLaj  is  also  called  the  impedance 
of  the  circuit.  It  evidently  has  for  magnitude  \/R2+L2a>2} 
and  as  an  operator  rotates  an  associated  quantity  counter- 
clockwise through  an  angle  6,  equal  to  tan"1  — . 

R 

It  is  perfectly  evident  that  R,  w  and  L  are  all  real 
quantities  and  that  R  and  LOJ  do  not  in  fact  have  any 
quarter-phase  relation.  To  express  the  proved  relation 
between  E  and  7  and  between  E  and  7,  we  merely  assume 
the  existence  of  a  physical  quantity  called  an  impedance, 
which  we  express  as  v/R2  +  L2oo2  or  R+jLto  in  the  two 
cases  respectively.  This  is  simply  a  case  of  the  end  justi- 
fying the  means.  It  is  evident,  however,  that  the  investi- 
gation of  the  relation  between  E  and  7  (either  effective 
or  maximum  values)  has  been  perfectly  general,  i.e.,  no- 
special  values  have  been  assumed  for  any  of  these  quantities. 
We  may  therefore  fearlessly  deal  with  impedances  just 
as  though  resistances  and  reactances  (as  we  designate 
products  like  Leo)  had  in  fact  perpendicular  (or  quarter- 
period  difference)  relations.  It  must,  however,  be  kept  in 
mind  that  we  are  considering  simple  harmonic  quantities 
and  that  for  other  quantities  other  results  follow. 

§  21.  Let  us  now  consider  the  case  of  a  simple  circuit 
with  simple  harmonic  e.m.f.  and  constant  resistance  R> 
inductance  L  and  capacity  C  all  in  series.  As  is  well 
known,  after  a  reasonable  time  the  current  reaches  its 
harmonic  state  and  may  be  expressed  as  before  by  the 
formula 

i  =  I  cos  (cut—  6). 

Ohm's  law  extended  to  variable  conditions  gives  for  the 
e.m.f., 

e  —  E  cos  oot  = 


28  REVOLVING  VECTORS 

Substituting  the  value  of  i,  we  have 

Jftcos  orf  =  /2  cos  (a)t-  6)  +  (Laj- 


K. 


It  is  evident  that  the  constant  K  must  be  zero,  otherwise 
the  second  member  of  the  equation  is  not  simple  harmonic. 
We  therefore  have 


FIG.  7. 


This  equation  is  illustrated  by  Fig.  7.     The  projections  of 
the  sides  of  the  triangle  on  the  horizontal  line  are  evidently 


Ecoswt 


—  6      and 


(  LOJ  —  ^-jcos(  a>t  —  #  +  ;H. 


The  triangle  is  understood  to  be  in  counter-clockwise 
rotation  about  the  point  0,  with  an  angular  velocity  co. 
From  the  triangle  it  is  evident  that 

1 


E 


and    tan  6  = 


R 


HARMONIC  CURRENT,  IMPEDANCE 


29 


The  reactance  of  the  circuit  is  Lco  —  -^-.     Incase  •^r 

Co)  Ca> 

the  angle  6  becomes  negative,  as  shown  by  the  diagram 
(Fig.  8),  but  the  form  of  the  equation  remains  unchanged. 
If  we  consider  the  projections  of  the  sides  of  the  triangles 
on  a  vertical  line,  we  have 

jE  sin  &>£  =  //  R  sin  (ajt  —  6)  +  (  L&>  — -^-  )  sin  I  cut—  #+H-)    • 


FIG.  8. 

Combining    these    two    equations   we    have,    remembering 
that  e^~= 


The      factor     R+jlLcj  —  -^-  j     and      its      magnitude 
\  J?2  +  f  La)  —  -~-  \    are  called  the  impedance  of  the  circuit. 


30  REVOLVING  VECTORS 

They  play  the  same  role  as  the  resistance  in  the  case  of 
unvarying  currents  for  which  Dr.  Ohm  formulated  his 
rule,  known  as  Ohm's  law.  This  factor  (in  either  form)  is 
the  ratio  between  simple  harmonic  e.m.f.  and  current, 
and  may  be  used  fearlessly  in  finding  the  value  of  the 
current  with  known  e.m.f.  and  vice  versa.  In  its  com- 
plex form  impedance  indicates  not  only  the  magnitude  of 
the  ratio,  but  also  the  fact  that  the  current  lags  behind 
the  e.m.f.  by  an  angle  6,  whose  tangent  is  expressed  by  the 
formula, 

Lo}  —  -^- 

0  LCO 

tan  0  =  -  -  . 


This  lag  becomes  a  leading  angle  if  Lco<-^—. 

Leo 


HARMONIC  ELECTROMOTIVE  FORCES  IN  SERIES 

§  22.  The  foregoing  method  may  be  used  for  circuits  in 
which  two  electromotive  forces  of  different  phase  are  in 
series  with  a  known  impedance,  or  for  divided  circuits  in 
which  the  current  in  different  branches  has  different  phases. 

As  an  example  of  the  former,  suppose  a  circuit,  of 
resistance  R  and  reactance  Leo,  includes  two  electromotive 
forces  in  series,  the  second  lagging  behind  the  first  by  a 
phase  difference  6.  It  is  assumed  that  the  frequency  is 
the  same  for  both.  The  combined  e.m.f.  expressed  in  a 
uniform  circular  formula  is 

E  =  E!  +  E2  =  fi*%  +  €««*-  »E2  =  e'**[Ei  +  e  ~  '  %] 
=  £'w[#i  +  (cos  0-/sin  0)E2], 
=  e'^Ei  +E2  cos  0-jE2  sin  0]. 
Let  us  assume  that  tan  6f  =        2    z,  and  derive  corre- 


HARMONIC  ELECTROMOTIVE  FORCES  IN  SERIES   31 
spending  values  for  sin  Q'  and  cos  6' '.     We  then  shall  have 


0'—  /sin  6' 


cos  0 


and 


cos    , 


cos    . 


The  impedance  of  the  circuit  is  R  +jLa),  and  it  produces 
a  lag  6"  in  the   current  behind  the  combined  e.m.f.     The 

value  of  6"  is  tan"1-^.     We  have  also  the  relation 
H 


R  +  jLu)  =  £Je"VR2  +  L2aj2. 
Therefore  the  current  is 

7=      E 


Example.  Assume  ^i  =  1000  volts,  ^2  =  1200  volts, 
6>  =  60°,  R=12  ohms,  Lw  =  5  ohms,  to  find  6',  6",  I,  and  E. 

Answers.  6' =  i&irl  0.6495,  0"  =  tan"1 -ft,  7=  146.76  amp., 
^=1907.9  volts. 

The  diagram  (Fig.  9)  indicates  graphically  the  mag- 
nitudes and  phase  relations  of  E\,  E2  and  E  and  the  phase 


32 


REVOLVING  VECTORS 


relation  of  /.  The  current  is  plotted  to  a  different  scale, 
however,  to  avoid  confusion  in  the  diagram.  The  whole 
diagram  is  understood  to  rotate  counter-clockwise  about 
0  with  an  angular  velocity  CD  =  2nf,  when  /  is  the  frequency 
(cycles  per  second)  of  the  e.m.f.'s. 

PROBLEM  OF  A  DIVIDED  CIRCUIT 

§  23.  The  problem  of  a  divided  circuit  is  as  follows: 
Assume  an  e.m.f.  E  between  junction  points  of  a  divided 
circuit,  in  one  branch  of  the  circuit  a  resistance  RI,  a 
reactance  L\w,  and  a  current  I\,  in  the  other  branch  similar 


FIG.  10. 

quantities  R2,  L2a>  and  72,  and  a  current  7  in  the  undivided 
part  of  the  circuit.  The  arrangement  of  the  circuit  is  as 
shown  in  the  diagram  of  connections  (Fig.  10).  Let  us 
take  as  known  quantities,  I\,  RI,  L\OJ,  R2  and  L2a>,  and 
let  us  find  E,  I2  and  7,  together  with  the  phase  relations. 

Let  us  write  ^1  =  tan~1  —  ^    and     ^2  =  tan~1  ~^-. 
H\  ri2 

Let  us  assume  the  phase  of  7i    to    be    the    standard. 
We  have  then 


7i  =  (cos  wt  +j  sin  wt)Ii  = 


PROBLEM   OF  A  DIVIDED  CIRCUIT  33 

and 


E  is  ahead  of  I\  in  phase  by  the  angle  6\.     In  a  similar 
way  we  have 


and 


The  phase  of  72  is  behind  that  of  E  by  the  angle  02.  Com- 
bining the  equations  for  E,  we  obtain  the  relation  between 
72  and  1 1  as  follows: 


2aj< 


where  A   is  written  for   the   expression     \/    *  ~^    l  ^ 

V2         22' 
and 


I2  is  ahead  of  /i  in  phase  by  the  angle  d\  —  62.  The 
whole  current  7,  that  is  the  current  in  the  undivided 
portion  of  the  circuit,  is 


Separating  real  and  imaginary  parts  of  7,  we  have 

7  =  7!(l+Acos  (6l  -  62)  +  JA  sin  (#i-#2)). 

Writing 

A  sin  (0i-02) 
^3  =  tan      l+AcoB(01-02)' 

and  deriving  the  values  of  sin  63  and  cos  #3,  we  obtain 


7  =  ^(cos  63  +  /  sin  63)  Vl+A2  +  2  A  cos  (0!  -  02)  , 
or 


7  =  fij  */!  Vl  +  A2  +  2A  cos  (0i  -02)  , 
and 

I=I1V-1+A2  +  2A  cos  (0i  -02). 


34 


REVOLVING  VECTORS 


The  phase  of  I  leads  the  phase  of  I\  by  the  angle 
61  —  63.  It  is  evident  that  precisely  the  same  equations 
would  have  been  reached  if,  instead  of  assuming  7i  to  be 
known,  we  had  assumed  knowledge  of  E,  I2  or  7.  The 
equations  would  simply  have  been  derived  in  a  different 
order. 

Example.  Assume  /i  =  100  amperes,  #i  =  5  ohms, 
1/1^  =  2.5  ohms,  R2  =  15  ohms,  Z/i&>=15  ohms,  to  find 
E}  I2  and  7  and  the  phase  relations. 

Answers.  #  =  559.0,  72  =  26.352,  7=125.28,  0i=  tan"1  J, 
02  =  tan"1 1=45°,  ^tan"1  0.066708  =  3°  49'  18". 


FIG.  11. 

§  24.  The  problem  of  a  divided  circuit  is  illustrated 
graphically  in  the  diagrams  (Figs.  11  and  12).  The  diagrams 
illustrate  a  problem  in  which  the  resistance  of  the  first 
branch  is  three  times  the  reactance,  while  in  the  second 
branch  resistance  and  reactance  are  equal.  From  the 
former  diagram  (Fig.  11)  the  e.m.f.  E  may  be  determined 
in  terms  of  RI,  LIOO,  and  72.  After  "E  is  determined,  the 
diagram  furnishes  the  means  of  finding  R2I2}  and  the  value 
of  72  may  be  determined.  The  latter  diagram  (Fig.  12) 
shows  the  relation  of  7i,  72  and  7.  In  this  I\  is  drawn 
parallel  to  R\Ii  of  the  other  diagram,  and  in  the  same  way 
I2  is  parallel  to  R2I2.  The  diagonal  represents  7  to  the 
same  scale  as  I\  and  72.  If  the  diagrams  are  drawn  care- 


RESOLUTION  INTO  COMPONENTS 


35 


fully  to  scale  they  are  an  excellent  check  on  the  accuracy 
of  the  analytical  solution,  though  it  is  evident  that  the 
analytical  method  must  be  more  accurate  if  carried  through 
without  error. 

In  both  the  case  of  series  circuits  and  that  of  divided 
circuits  it  is  possible  that  the  quantities  to  be  added  may 
differ  in  phase  by  large  angles  and  the  total  (so  called) 
of  the  e.m.f.  or  current,  in  the  different  cases  respectively 
may  be  less  than  either  component.  This  is  the  case  in 
a  series  circuit  including  a  motor  and  a  generator,  which 


FIG.  12. 


in  practical  cases  are  nearly  opposite  in  phase,  or  with 
condensers  and  inductances  in  series.  This  is  true  also 
of  currents  in  a  divided  circuit,  one  branch  having  inductance 
and  the  other  capacity. 


RESOLUTION  INTO   COMPONENTS 

§  25.  Instead  of  indicating  the  e.m.f.  and  current  as 
projections  of  uniform  circular  quantities  expressed  in  mag- 
nitude and  phase  (the  latter  as  a  function  of  the  time), 
we  may  express  the  circular  quantities  at  the  particular 
instant  in  terms  of  their  real  and  imaginary  components. 
Let  Ei+jE2  represent  the  circular  quantity  whose 
magnitude  \/Ei2+E22  is  the  effective  value  of  the  e.m.f. 
The  instantaneous  value  of  the  e.m.f.  is  \/2Ei,  as  explained 
elsewhere  (§19).  The  expression  /i+//2  in  the  same  way 


36  REVOLVING  VECTORS 

expresses     the    current    (effective    value  =  \//i2+/22,    and 
instantaneous  value  V2/i).     We  then  shall  have 


As  the  two  circular  quantities  are  equal  in  both  magnitude 
and  direction,  we  have  the  two  equations, 


and 

E2  =  Rl2  +  I  LCD  —  77- 


The  quantities  Ei}  E2,  /i,  /2,  rnay  be  positive  or  negative, 
or  some  may  be  positive  and  the  others  negative. 
We  also  have 


E2R-E1 


(Lco-7r-\ 
\          Cft;/ 


2     > 


and  this  is  equivalent  to  the  two  equations, 
7  2\   '       CV 

~~7 —    ~~i 

\   (       d 

and 


RESOLUTION  INTO  COMPONENTS  37 

Lastly  we  have 


and  from  this  it  follows  that 

r?_ 
" 

and 

T         1  _ 

~ 


Should  the  circuit  be  non-inductive,  we  shall  have 
L  =  0  with  corresponding  changes  in  the  formulae.  If  there 
is  no  capacity  in  the  circuit,  we  must  not  assume  C  equal 
to  zero;  for  a  condenser  with  zero  capacity  means  an 
open  circuit.  We  must  instead  simply  remove  the  term 

—  .  It  is  interesting  to  note  that  this  conies  to  the  same 
Caj 

result  as  if  the  capacity  had  become  infinite;  in  which 
case  a  finite  charge  (time  integral  of  the  current)  would 
not  cause  an  appreciable  potential  difference  between  con- 
denser terminals.  That  is,  the  condenser  will  not  inter- 
pose any  direct  or  counter  e.m.f.  in  the  circuit. 

§  26.  The  problem  of  series  circuits  with  more  than  one 
e.m.f.,  resistance,  inductance,  and  capacity  is  to  be  solved 

by  using  S#i,  2E2,  Sfl,  SLo;,  and  S-=-  in  place  of  the 

Co) 

single  quantities. 

The  problem  of  divided  circuits  is  treated  in  an  anal- 
ogous way  to  the  divided  circuit  problem  (§§  23,  24),  which 

we  have  already  considered,  by  substituting  Z/i&>  --  in 


place  of  LIU  for  the  first  branch  and  substituting  cor- 
responding expressions  for  L2a)}  etc.,  in  the  other  branches. 


38 


REVOLVING  VECTORS 


USE  OF  A  SYMMETRICAL  PAIR  OF  TRIANGLES 

§  27.  To  represent  the  e.m.f.  by  a  pair  of  uniform  cir- 
cular motions,  in  terms  of  the  current,  resistance,  and 
reactance,  as  expressed  by  the  equation 


FIG.  13. 

we  may  make  use  of  two  triangles  revolving  in  opposite 
sense  with  angular  velocities  a>  and  always  symmetrical 
with  respect  to  the  horizontal  line  (Fig.  13).  While  this 
mode  of  representing  a  simple  harmonic  quantity  is  com- 
plete and  has  no  parts  to  be  rejected,  it  is  evidently  too 
complicated  for  general  acceptance  and  use  by  engineers. 


CHAPTER  V 
PRODUCT  OF  TWO  HARMONIC  QUANTITIES 

§  28.  Let  us  now  consider  the  product  of  harmonic 
quantities  and  in  particular  the  power  of  an  electric  cir- 
cuit, the  product  of  current  and  electromotive  force.  It 
will  be  seen  in  general  that  the  product  of  two  simple 
harmonic  quantities  is  not  simple  harmonic,  but  compound 
harmonic.  In  the  particular  case  of  greatest  interest  to  us, 
that  is  electric  power,  the  product  may  be  resolved  into  a 
constant  plus  a  simple  harmonic  quantity  of  double  fre- 
quency. It  has  been  shown  earlier  that  if  two  complex 
expressions  be  multiplied,  the  product  will  be  complex. 
For  example  the  product  of 

J.  =  (cos  #  +  /sin  6)R     and     5=  (cos  </>+j  sin</>)$ 
is 

A-  B=  (cos  (#+<£)+/  sin  (6+<f>))R-S. 

In  general  the  rule  of  multiplication  is  as  follows:  The 
product  has  a  magnitude  equal  to  the  product  of  the  mag- 
nitudes of  the  factors,  and  makes  an  angle  with  the  axis 
of  reals  equal  to  the  sum  of  the  angles  made  by  the  factors 
with  that  axis.  If  the  factors  are  uniform  circular  in 
form  and  functions  of  the  time  t  and  angular  velocities 
w\  and  oj2  as  follows: 


A  =  (cos  coit+j  sin 
B=  (cos  (jj^t  +/  sin 


39 


40  REVOLVING  VECTORS 

the  product  is 

A  -B=  (cos  (a>i+a>2)t+j  sin  (a)i+a>2)t)R-S, 

a  result  which  indicates  that  the  product  has  a  magnitude 
equal  to  the  product  of  the  magnitudes  of  the  factors  and 
an  angular  velocity  equal  to  the  sum  of  those  of  the  factors. 
If  a>i  equals  w2,  the  product  has  an  angular  velocity  twice 
as  great  as  the  factors. 

§  29.  For  the  sake  of  a  reductio  ad  absurdum  let  us 
assume  that  the  power  of  an  electric  circuit  may  be  obtained 
from  such  a  product  of  two  uniform  circular  expressions 
for  current  and  e.m.f.  as  follows: 

7=  (cos  (a)t-0)+jsm  (a)t-6))I, 
E  =  (cos  <t)t  +  j  sin  a)t)Et 


and 

EI  =  (cos(2cot-  6)  +j  sin  (2cot-  0))EI. 

The  product  El  is  a  uniform  circular  quantity  of  double 
frequency,  as  shown  by  the  factor  (2a>t—6),  and  has  an 
average  value  zero  for  its  projection  on  the  axis  of  reals. 
This  product  evidently  is  not  power,  for  the  power  of  an 
electric  circuit  has  in  general  an  average  value  different 
from  zero. 


POWER  IN  SIMPLE  CIRCUITS 

§  30.  Let  us  now  take  the  similar  simple  harmonic 
expressions  for  current  and  e.m.f.  in  terms  of  effective 
values  of  I  and  E: 

cos   tot-6=--I£i(«t-(»  +  £-X»t-V 


\/2 
cos  a)t  =  - 


POWER  IN  SIMPLE  CIRCUITS 


41 


By  multiplication  we  obtain  the  power  p,  with  average 
value  P, 


=  ei  =  2EI  cos 


(cot—d) 

77*  T 


(2cot-6), 


cos  6 


cos  (2a>t-6). 


This  expression  shows  that  the  instantaneous  value  of 
the  power  is  equal  to  a  constant  P  plus  a  simple  har- 
p 

monic  quantity  „  cos  (2ajt—6)   of  twice  the  frequency 

cosy 

of  the  current  and  the  e.m.f.  This  may  be  expressed  in 
circular  form  provided  the  origin  0'  be  taken  eccentric 
to  the  circle. 


FIG.  14. 


The  diagram  (Fig.  14)  expresses  the  power  in  circular 
form.  The  instantaneous  value,  p,  of  the  power  is  expressed 
by  the  distance  and  sense  of  O'Q.  The  maximum  value 


42  REVOLVING  VECTORS 

of  the  power  is  0'Q\,  and  the  minimum  (negative  maxi- 
mum) is  O'Qz.  As  P  cannot  be  greater  than  El,  and  may 
only  equal  El  when  6  is  zero,  the  point  0'  must  not  be 
exterior  to  the  circle.  The  circular  formula  is  as  follows: 

P  =  P  +  [E/]  =  P  +  (cos  (2cot-6)+jsm  (2a>t-0))EI, 

where  P  denotes  an  eccentric  uniform  circular  quantity 
made  up  of  a  constant  P  and  the  concentric  uniform  cir- 
cular quantity  [El]. 

§  31.  It  is  interesting  to  see  how  Dr.  Steinmetz  by 
introducing  an  arbitrary  method  of  multiplication  is  able 
to  obtain  the  average  value  of  the  power  from  the  circular 
f  ormulsB  (concentric)  for  E  and  7.  He  says  1  : 

"For  the  double  frequency  vector  P,  j2=  +  1,  or 
360°  rotation  and  j'Xl=/  and  lX/=—  /.     That  is, 
multiplication  by  /  reverses  the  sign/'   .  .   . 
Applying  his  rule  we  obtain  the  correct  result  as  follows: 
E=  (cos  cot+j  sin  a>£)E, 
7=  (cos  (w*-0)+/sin  (ajt-0))I, 
JEJ/  =  [cos  cot  -cos  (cot—  6)  -fsin  cot  sin  (cot—  6) 

+/(sin  cot  -cos  (cot—  0)  —  cos  6j£-sin  (cot—0))]EI, 


The  real  component  of  the  product  El  is  the  power 
El  cos  6  (average  value)  ,  the  imaginary  component  is  the 
so-called  wattless  component  of  the  power  El  sin  6  in 
magnitude. 

It  may  be  urged  in  objection  to  this  method  of  obtaining 
power,  that  while  the  real  part  of  the  product  is  the 
average  value  of  the  power,  we  must  take  the  whole  mag- 

1  Steinmetz,  Alternating  Current  Phenomena,  3d  edition  p.  151. 


POWER  IN  SIMPLE   CIRCUITS  43 

nitude  of  the  factors  for  effective  values  of  the  e.m.f.  and 
current,  and  the  real  parts  of  these  factors  must  be  mul- 
tiplied by  \/2  to  give  the  instantaneous  values  of  these 
quantities. 

§  32.  Another  arbitrary  method  of  combining  e.m.f.  and 
current  to  obtain  power  is  as  follows:  Let  the  e.m.f.  and 
current  be  represented  in  effective  value  by  the  magnitudes 
of  the  complex  quantities, 

E  =  EI  +  jE2  =  (cos  cot + j  sin  cot)  E, 

/  =  /i+//2=(cos  (cot- 6) +j  sin  (cot-d))I. 
We  know  from  previous  proof  that  the  average  value 

Tjl 

of    the    power  P  is   P  =  EI  cos  6,     also     that  -=f-=cos  otf, 

zi 

T?  J  T 

-rr  =  sin^,  y  =  cos  (cot -6),  y  =  sin  (cot -6)   and  that 

cos  0  =  cos  (cot}  cos  (cot  —0)+  sin  cot  •  sin  (cot  —  0) . 
It  therefore  follows  that 

EJi  +E2I2  =  EI  cos  6  (a  constant). 

As  El  cos  6  has  a  constant  value,  although  EI,  E2) 
/i  and  72  are  all  variables,  it  is  evident  that  we  shall  obtain 
the  correct  result  for  average  value  of  the  power  if  we 
take  the  values  of  these  variables  at  any  one  time. 

We  therefore  have  as  an  arbitrary  rule  to  obtain  the 
average  value  of  the  power:  multiply  the  real  parts  of 
e.m.f.  and  current,  and  the  imaginary  parts,  ignoring  the 
j2  and  add  the  products. 

Thus 

E=El+jE2, 

J-/1+//1, 

P=E1Il+E2I2. 


44  REVOLVING  VECTORS 

If  a  minus  sign  is  expressed  in  one  of  the  factors  it  must 
not  be  ignored.     For  example  if  we  have 

E=El-jE2) 

J-/i+/fc, 

then 

P=E\II  —  E2i2. 

If  we   have   a   circuit  including  a  motor  whose  e.m.f.   in 
general  opposes  the  current,  we  have  for  example 


P=-E1I1+E2I2. 

If  we  have  two  minus  signs  they  must  both  be  regarded, 
for  example 

E=E1-jE2, 


P=  EJi+  E2I2j 
etc. 

It  is  evident  that  the  above  examples  are  not  examples 
of  real  multiplication  of  current  and  e.m.f.  They  are 
merely  the  expression  of  a  rule  for  finding  average  power 
where  the  e.m.f.  and  current  are  known  in  magnitude  and 
phase  relation. 

§  33.  The  above  process  is  not  easily  reversed;  for  if 

P=E1I1+E2I2, 

and  supposing  I\  and  I2  known,  it  is  evident  that  there 
are  an  indefinite  number  of  values  of  EI  (with  corresponding 
values  of  E2)  which  will  satisfy  the  equation.  The  correct 
values  of  EI  and  E2  can  only  be  found  when  more  data 


POWER  IN  CIRCUITS  OF  MORE  THAN  ONE  PHASE  45 

are  given.     It  suffices  when  6  is  known.     We  have  had 
the  known  relation  P=EI  cos  $,  and  it  may  be  shown  that 

E  =p7i-72tan  6 
and 


In  these  formulae  6  is  the  angle  of  lag  of  the  current 
behind  the  e.m.f.  If  6  is  taken  as  the  angle  of  lead,  the 
terms  involving  tan  6  must  be  altered  accordingly. 

It  is  on  the  whole  more  satisfactory  to  use  another 
method  and  find  directly  from  the  relations, 

P  =  EI  cos  6, 
and 

I=I(cos  (ajt-6)+jsm  (cot-d)), 
the  result, 

p 

E  = —  (cos  cut  +  i  sin  cot] . 

•      I  cos  0 


POWER  IN  CIRCUITS  OF  MORE  THAN  ONE  PHASE 

§  34.  The  power  in  circuits  of  more  than  one  phase 
may  be  obtained  by  simple  addition,  whether  we  are 
dealing  with  instantaneous  or  average  values.  An  inter- 
esting case  is  that  of  balanced  circuits  having  two,  three 
or  more  phases ;  for  in  every  case  of  circuits  of  more  than 
one  phase  the  power  is  constant  during  the  whole  cycle 
if  all  the  currents  and  e.m.f. 's  of  the  various  phases  have 
equal  effective  values  respectively,  and  if  the  phase  dif- 

360° 

ferences  from  each  to  the  next  equals ,  where  n  is  the 

n 

number    of    phases.     The    two-phase    circuit    follows    the 
same    rule,    though    it    does    not    come    under   the    above 


46  REVOLVING  VECTORS 

statement  of  phase  difference;   for  in  it  we  have  one  inter- 
val of  90°  and  the  next  270°. 


BALANCED  TWO-PHASE   CIRCUIT 

§  35.  Let  us  take  the  balanced  two-phase  circuit  first. 
It  has  been  shown  that  the  power  at  any  instant  in  an 
alternating  current  circuit,  with  simple  harmonic  e.m.f. 
and  current  is 

p  =  EI(cos  d  +  cos  (2a)t-6)). 

If  the  current  and  e.m.f.  in  the  second  branch  of  the 
circuit  have  the  same  effective  values  as  in  the  first,  but  a 

phase  difference  of  —    the  variable  part  of  the  power  will 

Zt 

have  a  phase  difference  of  TZ,  because  power  is  a  double 
frequency  variable.  We  therefore  shall  have  for  p\  and 
p2,  the  power  of  the  two  phases, 

pi  =  EI(cos  6  +  cos  (2a>t-6)), 

p2  =  EI(cos  6-cos  (2ajt-0)), 
and 

p  =  pi  -j-  p2  =  2EI  cos  0  =  Pt     (a  constant) . 


BALANCED  THREE-PHASE  CIRCUIT 

§  36.  For  a  balanced  three-phase  circuit  we  shall  have, 

27T 

if  the  phase  intervals  are  120°  =  —, 

o 

Pl  =  El  (cos  6  + cos  (20)1-6)), 
p2  =  El  (cos  6  + cos  (2ajt-6+$n)) 

=  EI(cosd  +  cos 
p3  =  El  (cos  0  +  cos  (2a>t  -  6  +  f  TT)  ) 

=  EI (cos  0  +  cos 
and 

cos6  =  P. 


BALANCED  FOUR-PHASE  CIRCUIT 

It  is  evident  that  the  variable  parts  of  p  annul  one 
another,  for  they  may  be  represented  as  the  projections 
of  the  three  sides  of  an  equilateral  triangle,  each  side 
equal  to  El  in  magnitude  and  the  first  side  making  an 
angle  with  the  line  on  which  they  are  projected  equal  to 
2wt—6.  It  is  well  known  that  the  sum  of  the  projections 
of  the  sides  of  any  closed  polygon  is  zero.  This  relation 
is  illustrated  by  the  diagram  (Fig,  15), 


FIG.  15. 


BALANCED   FOUR-PHASE   CIRCUIT 

§  37.  The  four-phase  case  is,  in  a  similar  way,  shown 
to  have  constant  power.  Or  it  may  be  looked  upon  as  two 
pairs  of  balanced  two-phase,  each  pair  as  shown  above 
having  constant  powei.  Therefore  the  sum  of  all  four  has 
constant  power. 


48 


REVOLVING  VECTORS 


BALANCED   SIX-PHASE   CIRCUIT 

§  38.  The  six-phase  case  is  evidently  a  case  of  two 
balanced  three-phase  systems.  The  variable  part  may  be 
represented  by  the  projections  of  the  sides  of  an  equilateral 
triangle  each  being  taken  twice.  The  power  of  the  parts 
and  the  whole  are 

pi  =  El  (cos  6  +  cos 
p2  =  El  (cos  6  +  cos 
p3  =  El  (cos  6  +  cos 

P4  =  El  (COS  6  +  COS 

p5  =  EI(cosd  +  cos 
pQ  =  El  (cos  6  +  cos 


=  COS 


BALANCED   POLYPHASE    CIRCUITS    IN   GENERAL 

§  39.  If  there  are  an  odd  number  of  phases,  more  than 
three,  the  variable  part  of  the  power  is  represented  by 


FIG.  16. 

the  projections  of  the  lines  of  a  star-shaped  diagram  which 
is  in  all  cases  a  closed  figure  with  a  total  of  zero  for  the 
projections.  The  five-phase  case  will  suffice  for  illustra- 
tion (Fig.  16).  For  five  phases  the  progressive  interval 


BALANCED  POLYPHASE  CIRCUITS  IN  GENERAL     49 

is  IT:  =  72°  for  current  and  e.m.f.  and  £7r=1440  for  the 
variable  part  of  the  power.  The  power  of  the  various 
phases  and  the  total  are  as  follows: 

0  +  cos  (2wt-6)), 
0  +  cos  (2ajt- 


=  El  (cos  0+cos  (2otf-0- 

7i)) 
=  £7  (cos  fl  +  cos  (26>*- 


=  EI(cos  6  +  cos  (2tot- 

=  EI(cos 


COS      = 


The    eccentric    circular    diagram    for    the    power    of    a 
balanced  polyphase  system  of  n  phases  reduces  evidently 


FJG.  17. 

to  a  horizontal  line  of  length  nEI  cos  6  and  a  closed  regular 

polygon  of  —  sides  with  each  side  taken  twice  if  n  is  even, 

2t 

or  a  regular  star  of  n  sides  if  n  is  odd.     The  diagram  (Fig. 
17)   illustrates  an  eight-phase  balanced  system,   in  which 


50 


REVOLVING  VECTORS 


the  instantaneous  power  is  a  constant  quantity  P.  The 
variable  part  of  the  power  of  each  separate  phase  is  the 
projection  of  the  corresponding  side  of  the  square,  each 
being  taken  twice  as  indicated. 

In  the  same  way  the  diagram  (Fig.  18)  represents  the 
power   of  a  five-phase  balanced  system.      As  before,   the 


O  P 

FIG.  18. 

instantaneous  power  is  constant  for  the  system,  the  variable 
part  of  the  power  of  the  separate  phases  being  represented 
by  the  projections  of  the  lines  of  the  five-pointed  star. 

UNBALANCED  POLYPHASE   CIRCUITS 

§  40.  While  in  general  we  expect  the  power  to  be  con- 
stant only  in  balanced  symmetrical  polyphase  (or  two-  or 


FIG.  19. 


three-phase)   systems,  it  is  evident  that  any  system  will 
have    constant   power   if   the    eccentric    circular   diagram 


UNBALANCED  POLYPHASE  CIRCUITS  51 

gives  a  closed  figure  for  the  variable  parts  of  the  separate 
phases.  As  a  rule  unbalanced  polyphase  systems  do  not 
give  a  closed  figure  for  the  variable  parts  of  the  eccentric 
circular  diagram. 

The  diagram  (Fig.  19)  illustrates  the  power  of  an  un- 
balanced three-phase  system  for  which  P,  the  average 
value  of  the  power,  is  represented  by  00\. 

The  maximum  value  of  p  is  002  and  the  minimum 
is  003. 


CHAPTER    VI 
NON-HARMONIC   CURRENTS 

§  41.  The  method  of  revolving  vectors  has  interesting 
applications  to  cases  of  currents  which  are  not  harmonic 
in  the  strict  sense  of  the  word.  The  cases  which  we  shall 
here  investigate  are,  first,  the  oscillatory  discharge  of  a  con- 
denser, second,  its  non-oscillatory  discharge  and  third,  the 
current  following  the  closing  of  a  circuit  in  which  the  e.m.f. 
is  simple  harmonic.  In  this  last  case  it  is  well  known  that 
the  current  is  not  harmonic,  but  as  time  goes  on  approaches 
more  and  more  nearly  to  harmonic  values. 

OSCILLATORY  DISCHARGE   OF   A   CONDENSER 

§  42.  First  let  us  consider  the  oscillatory  discharge  of 
a  condenser.  Let  the  capacity  of  the  condenser  be  repre- 
sented by  C,  the  e.m.f.  to  which  it  is  charged  by  E0  when 
the  circuit  is  about  to  be  closed,  and  by  e  at  later  times. 
Let  the  current  be  indicated  by  i,  the  resistance  of  the 
circuit  by  R  and  the  inductance  by  L.  It  will  be  assumed 
that  C,  R,  and  L  are  all  constants.  We  shall  have  as 
the  form  of  Ohm's  law  applicable  to  variable  conditions 


fidt 

-- 


or 


OSCILLATORY  DISCHARGE  OF  A  CONDENSER       53 

Differentiating  this    expression  we  have,   after  rearranging 
the   terms 


This  equation  has  two  solutions,  each  of  the  form 


where  K  and  a  are  constants  to  be  determined.  The  con- 
stant K  depends  on  EQ,  as  will  be  shown  later,  and  cannot 
be  found  from  the  differential  equation  in  its  latter  form. 
The  values  of  a  are,  however,  to  be  found;  for  substituting 
i  =  Ks~a*,  we  obtain 


We  shall  assume  that  the  current  i,  which  equals  Ks~at} 
is  not  zero  in  general.     We  therefore  have 


LC" 

and 


R2       1       «/- 1      4L 


or 


_R      JR2       1       R 

GL  —  ,-.  T 


The  general  solution  for  i  may  include  both  values  for 
a  and  different  values  for  K  which  we  shall  designate  by 
KI  and  K2.  The  equation  for  the  current  is  then 


54  REVOLVING  VECTORS 

As  it  is  evident  that  at  the  time  t  of  closing  the  circuit, 


=  0  and  i  =  0}  we  shall  have  therefore 


or 
and 


If  a\  and  a2  are  real  quantities,  an  investigation  of 
this  equation  will  show  that  the  current  will  start  at  a 
value  zero,  rise  to  a  maximum,  and  fall  off  later  to  smaller 
and  smaller  values  without  reversal  of  sign,  and  become 
zero  only  after  an  infinite  time  has  elapsed.  This  condition 
is  expressed  by  the  inequality  R2C>4L.  If  on  the  other 
hand  R2C<4L,  it  is  evident  that  a\  and  a2  are  complex 
quantities  as  follows: 


R      .R  ^L          _  R 
2L+12L*R2C       ~2L 


R 

2L 

where  we  have  written 


Converting  the  exponentials  with  imaginary  indices  into 
sine  and  cosine  terms,  and  remembering  that  in  this  par- 
ticular case  the  current  must  be  zero  when  t  is  zero,  we 
have 


The  period  of  the  oscillation  of  the  discharge  (or  cur- 
rent) is 


OSCILLATORY  DISCHARGE  OF  A  CONDENSER       55 

If  R2  is  small  in  comparison  with  -^,  the  period  is 

C 

T  =  27zVLC     (approximately)  . 

With  larger  values  of  R  the  period  is  increased  until 
when  R  equals  2  /—  ,  the  period  becomes  infinite.  If  R2 

is  greater  than  -^  the    discharge  is  aperiodic  (without    a 
u 

period)  ,  corresponding  to  the  earlier  formula. 

Let  us  now  investigate  e,  the  e.m.f.  which  equals  the 
potential  difference  between  condenser  terminals.  We  have 
the  relation 


Substituting  the  value  of  i  as  given  above,  we  have 

_Af  f  R          1 

e  =  Ks    2L     #sin^  +  L/?cos/ft--sin/?n 


Writing,  to  simplify  the  expression,  tan  £=      pvr^—  1, 
we  have 


As  on  closing  the  circuit  (i.ev  i  =  0)  we  had  E0  =  e,  we 
obtain  as  the  value  of  the  constant  K, 


56  REVOLVING  VECTORS 

and 
e  = 


V4L-R2C 
and 


4<  .    IRt    /4L  .    /4L       A 

2  sm(2i\te^~ 1+ tan  V^~  /' 


2£0 VC      -^«  .    /#£    [4L      "\ 

— ~ c-      ^^     em  I  I \ 

—       /  — ft  bill  I  _  T  ^  /  ,-jO/-v       J.  I . 

>/4L-i22C  \2L\R2C      ) 

If  7^2  is  small  in  comparison  with  -^ ,  the  current  for  early 

0 

oscillations  is 

1C  l 

i  =  E0.  -  sin  ~7==      (approximately) . 

\/  />          'v  I A  j 


The  maximum  value  of  the  current  (complete  expres- 
sion) occurs  in  less  than  one-quarter  of  a  period  after 
closing  the  circuit,  at  a  time  when 


as  may  be  shown  by  putting  -r-  equal  to  zero.  The  mini- 
mum value  of  i  occurs  half  a  period  later,  when  the  sine 
equals—  ./I  —  -TJ-,  &  result  also  of  the  same  equation,  -^-=  0. 

Using  the  notation  of  rotating  vectors  we  may  express 
e  and  i  as  the  real  part  of  two  exponential  spirals  E  and  / 

s  s 

as  follows: 


VlL-RtC 


OSCILLATORY  DISCHARGE  OF  A  CONDENSER        57 
The  charge  of  the  condenser  is  evidently 


This  may  be  written  as  the  real  part  of  the  exponential 
spiral 

2E0CVL     -+ 


V4L-R2C 


§  43.  The  results  for  i,  e,  and  q  might  bave  been  deduced 
in  the  form  of  the  difference  of  two  exponential  spirals, 
directly  from  the  solution, 


for  if  a\  and  a^  are  complex  both  KIS  ait  and  K^e  a2t  are 
exponential  spirals.  If  the  result  is  to  be  real  at  every 
instant,  it  is  evident  that  the  imaginary  parts  of  both  spirals 
must  be  equal,  while  the  real  parts  must  be  equal  in  mag- 
nitude but  of  opposite  sign.  It  is  evident  that  both  spirals 
must  start  for  t  =  Q  at  a  point  for  which  the  real  values 
are  zero.  To  satisfy  this  condition  K\  must  be  a  pure 
imaginary.  As  we  shall  prefer  to  keep  the  constant  real, 
we  may  reach  the  same  result  by  changing  the  phase  of 

the  spirals  by  an  angle  — .     Remembering  that  /2=/  and 

-i— 

s  2  =  —  /,  we  may  write  i  as  the  sum  of  two  exponential 
spirals,  or 


where  K  is  the  magnitude  of  the  pure  imaginary  KI  as 
explained  above. 


58  REVOLVING  VECTORS 

Substituting  the  value  of  i  in  the  equation  for  e, 


we  obtain 


or 


K\(R     .R      J        \  -«rf+/| 
*IU~W^     T 


R,JI  *L   :\  -«,f-#fi 

2+^V^C-T 


4L 


Writing  as  before  tand=    /7^-l,sin£=    /1-— -    etc., 


and  remembering  that  l±j     —--  1=    I^JLe±i8  we  have 

^  ^ 


after  rearranging  the  terms, 


At  the  time  of  closing  the  circuit  (i.e.,  ^=0),we  had 
therefore 


or 


and 


1E-2PC 


NON-OSCILLATORY  DISCHARGE  OF  A  CONDENSER    59 

Substituting  the   value   of  K  in  the   earlier   formulae,   we 
have 


Rt  f     •  /Rt     |~4l  * 

fl__  |    ,  (^          !_ 


V4L-R2C 

IL  o\f)  I .     /_, I   j 

\2L\Tt2C         ' 

\  if  / 


sin 


The  charge  of  the  condenser  has  a  similar  "formula,  derived 
from  the  relation,  q  =  eC.  It  is  unnecessary  to  write  it 
out  in  full. 

The  exponential  spirals  for  each  case  may  be  plotted 
in  polar  coordinates,  and  will  be  seen  to  be  in  all  respects 
equal -except  that  one  is  right  handed  and  the  other  left 
handed.  The  epoch  angles  for  the  starting  points  are 
indicated  in  the  exponent. 


NON-OSCILLATORY   DISCHARGE   OF   A   CONDENSER 

§  44.  It  has  been  shown  above,  §§42  and  43,  that  the 
condition,  R2C  <4L,  corresponds  to  an  oscillatory  discharge 
of  the  condenser.  If,  on  the  other  hand,  we  have  either 
R2C>4L  or  R2C=4:L,  then  the  discharge  will  be  non- 
oscillatory. 

The  general  solution 


60  REVOLVING  VECTORS 

in  which  «i  and  0.2  are  real,  corresponds  to  the  condition 
R2C>4L]  but  becomes  indeterminate  if  R2C=4:L,  or 
.ai  =  a2.  In  this  latter  case  we  must  resort  to  the  par- 
ticular solution, 

i=(Kl+K2t)e-«t. 

First,  let  us  consider  the  general  solution.     At  the  time 
of  closing  the  circuit,  we  have 


or 

K= 


Substituting  the  values  of  «i  and  0:2,  §  42,  we  have 

7?/ 

i=  °r     or         P2^ 


.RL/  ***-  i 


VR2C2-4LC 
If  R2C=4L,  we  have  from  the  particular  solution 

giving  as  a  final  result 


PHENOMENA    OBSERVED    ON  CLOSING  THE  CIRCUIT, 
STARTING  TERM 

§  45.  Let  us  now  consider  the  expression  for  the  cur- 
rent in  a  circuit  which  has  just  been  closed,  the  e.m.f. 
being  simple  harmonic.  As  is  well  known,  the  current 
will  not  become  harmonic  at  once,  even  though  the  elec- 
tromotive force  is  precisely  harmonic.  The  divergence 
from  harmonic  values  is  expressed  in  the  equation  for 
the  current  by  a  so-called  "  starting  term."  Let  us  con- 
sider a  circuit  of  constant  resistance  R,  inductance  L, 


PHENOMENA  OBSERVED  ON  CLOSING  THE  CIRCUIT    61 

capacity  C  (all  in  series)  and  a  simple  harmonic  e.m.f.  e, 
equal  to  E  cos  cot.  We  have  as  the  expression  for  Ohm's 
law  extended  to  variable  e.m.f. 


It  is  well  known  that  in  such  a  circuit  the  current 
eventually  will  follow  a  simple  harmonic  law.  Indicating 
the  starting  term  by  ^1,  <f>  being  a  function  of  the  time 
t,  and  I  being  the  maximum  value  of  the  current  after  the 
harmonic  condition  is  reached,  we  have 

j_!(qft*-  0)  +  e-j(*>t-  6))  _  ^lm 

Let  us  substitute  the  value  of  i  in  the  previous  equation. 
We  obtain 


The   constant   of  integration   is   reserved  for  the   last 
term.     As   no   particular   values   have   been   assumed   for 

i.-i 

I  or  0,  it  is  evident  that  if  we  write  tan  6=  -  ^  —  —  and 

H 

E2  =  I2  n2  +  (LOJ  -  -1-}  2\   we  shall  have 


(LOJ  -  -1-}  2\  , 


La>-i^*^ 


and  therefore 


62  REVOLVING  VECTORS 

Differentiating  the  last  equation  and  dividing  by  L, 
we  have 


Rdj      <[> 
dt2     L  dt  ^CL 

We  have   already  found  the  solution  of  an  equation  of 
this  form  (§  42).     If  kL>R2C  we  may  write  at  once 


4L 


where  K  and  7-  are  both  real  quantities.  This  is  equivalent 
to  dropping  the  7-  from  the  expression  and  giving  the  expo- 
nentials in  the  bracket  different  factors  KI  and  K2.  If 
the  latter  mode  of  expression  were  used,  KI  and  K2  would 
in  general  be  found  to  be  complex  constants. 
If  R2C>  4L  the  solution  for  <r  takes  the  form 


t 


Kn    2i(1  +  V'     «fe)+#2£    **b    V1"^), 


which  we  shall  discuss  later  in  §  48. 

§  46.  To  determine  the  values  of  K  and  f,  we  must 
know  the  condition  of  the  circuit  at  the  time  it  is  closed. 
As  we  have  already  assumed  in  our  formula  for  the  elec- 
tromotive force  that  e  is  at  its  largest  value  when  £=0, 
or  any  number  of  complete  periods  later,  we  cannot  in 
fairness  assume  that  the  time  of  closing  the  circuit  is 
necessarily  the  same.  Let  us  then  take  the  time  of  closing 
the  circuit  to  be  fo.  Let  us  assume  that  the  condenser  was 
already  charged  to  a  potential  E0  when  put  in  circuit. 
We  therefore  have  the  conditions  that 


i  =  0        and 
--=E  cos  u)t  — 


at  the  time,  to,  of  closing 
the  circuit. 


PHENOMENA  OBSERVED  ON  CLOSING  THE  CIRCUIT    63 

Indicating  by  00  the  value  of  ^  when  t=tQ,  we  evidently 
have  from  the  relations  above, 

0o  =  cos  ((otQ  —  6) 
and 


or 


E  cos  wto- 


[dd>~\  Ecoscot0—EQ 

iLr     -w-    ~ 

Writing,  as  in  §  42, 


4L     i 
~ 


the  equation  for  the  starting  term,  §  45,  becomes 
m  m 


and  when  t=t0  we  have 

Rto 


cos  =  cos  o>    -     . 

We  have  also 


Ecosa)t0—E0 
=  --        ,  j       — 


From  these  equations  we  obtain 

E  cos  ajto  —  E0  +  La)I  sin  (ajt0-6)       R 


64  REVOLVING  VECTORS 

and 


.     _!  f          2C          IE  cos  (jot0 —  E0  + Lcol  sin  (ojt0  —  0) 
L  V4£C-R2C2  \  I  cos  (w^0  -  6) 

R\\   Rt0    I  4L 
2/J    2LA/T^^ 
and 

X=4£^-ar/"'~/h 


JC  (2^'cos  wt0  -  2E0  +  2La>7  sin  O*0  -  ^) 



Depending  on  which  sign  is  taken  for  the  square  root, 
K  may  be  either  positive  or  negative.  It  is  simpler  to 
take  the  positive  value,  in  which  case  cos  (/ft0  +  r)  ig  positive 
also.  If,  however,  the  negative  value  of  K  is  chosen, 
cos  (ftt0  +  f)  is  negative  also,  with  a  consequent  change  of 
TT  in  the  value  of  f.  The  formula  for  f  has  the  correspond- 
ing ambiguity. 

It  is  possible  under  certain  circumstances  for  the  current 
to  be  harmonic  from  the  time  of  closing  the  circuit.  In 
this  case  the  starting  term  becomes  zero.  This  requires 
that  the  circuit  be  closed  at  the  instant  when  cos  (cot—  6) 
is  zero,  and  that  the  initial  potential  difference  between 
condenser  terminals  has  the  value 


E0  =  E  cos  cut0  +  LcijI  sin  (a>t0—6). 

§  47.  Graphically  the  current  may  be  represented  by 
the  resultant  of  four  revolving  vectors  made  up  of  two 
pairs.  The  first  pair  consists  of  two  uniform  circular 
vectors,  revolving  in  opposite  directions  with  angular 
velocity  a>  and  with  equal  magnitudes.  The  second  pair 


PHENOMENA  OBSERVED  ON  CLOSING  THE  CIRCUIT    65 

consists  of  two  exponential  spirals  with  angular  velocities 
-B^;—  1    in    opposite    directions    and    with    equal 


magnitudes. 

§  48.  If  the  resistance  of  the  circuit  is  greater  than  or 

equal  to  2    /—  ,  the  starting  term  loses  its  oscillatory  char- 
acter, and  the  formula  for  tp  becomes  in  the  former  case 


where 


S^-Jfe 


=  R   .  R    L      4L          ,  R      R    i      4L 

4 

and 


At  the  time,  t0,  of  closing  the  circuit,  the  current  is 
zero  and 


HI-*' 


E  cos  a)to—EQ. 


Substituting  the  values  at  the  time  t  =  t0,  we  obtain 
0  =  /[cos  (a)tQ-0)-<f>0]  =  I[co8  (a}tQ-6)-Kl£-a^^- 
E  cos  <tito-E0  =  LI-a)Bi 


Substituting  in  this  equation  the  value  from  the  previous 
equation  of  K2£-a*io,  K2£~a*to=cos  (a}t0-6)-Ki£-ait\  we 
have 

E  cos  ojtQ  —  Eo+LajI  sin  (wto—  6) 

cos      t-O 


66  REVOLVING  VECTORS 

and 

pr    -™t«—E  cos  toto-Eo+La>I  sin  (cot^—  0)  —  a2LI  cos  (a}t0— 


„    _    fo  __  E  cos  a)to—E0+LcoI  sin  (a>to—0)—aiLI  cos  (cot0—d) 

(ai-a^Ll 

and 

._  E  cos  a)to—E0+La>I  sin  (ajt0—d)  —a2LI  cos  (ajto—6)   _    .      . 
*=  (a1-a2)LI 

E  cos  &>£o—  EQ+LOJ!  sin  (cotQ—0}—a\LI  cos  (ajtQ—0)  _a  (<_n 
(a  i-az)Ll 

If  jK2C  equals  4L,  the  starting  term  takes  a  third  form, 


where  -SCi  and  ^"2  are  constants  depending  on  the  conditions 
at  the  time  tQ  of  closing  the  circuit.  Making  the  same 
assumptions  as  before,  we  shall  find  that 


and 

ffTJ?         .  Ecoscot0-E0          ....         "] 

2=         2L  C°S  ^  °~   ^  ~       TJ  Sm  ^  °~  ^    ' 

If  the  current  is  to  be  harmonic  from  the  time  of  closing 
the  circuit,  evidently  we  must  have  in  the  last  two  cases, 
as  in  the  first,  §  46, 

cos  (ojtQ-d)=0, 
and 

EQ=E  cos  (jL>tQ  +  La>I  sin  (cot0—6). 


GENERAL  REMARKS  67 


GENERAL  REMARKS 

§  49.  It  is  evident  also  that  these  results  for  the  oscil- 
latory discharge  of  a  condenser  and  for  the  current  with 
simple  harmonic  e.m.f.  when  the  starting  term  is  oscillatory, 
might  be  expressed  as  the  real  part  of  an  exponential 
spiral  for  the  first  case,  and  as  the  real  part  of  the  sum 
of  a  uniform  circular  quantity  and  an  exponential  spiral 
for  the  second. 

For  the  oscillatory  discharge  we  shall  have 


f     2E0VC      _»±,-/»   /J£l7_JLn 
i  =  real  part       .  =g    2L±J  \2L^  R*C    ]     2      , 

LV4L-R2C  J 


r  2E0        _»±,./»         -^+^l 

e=  real  part       .  =e    2L±J  \2L^  we        a*7  I, 

LV4L-R2C  J 

and  for  the  current  when  the  e.m.f.  is  simple  harmonic, 
[ 


{  =  real  part/ 

In  general  the  angular  velocity  /?  of  the  spiral  will  be 
different  from  the  angular  velocity  o>  of  the  uniform  cir- 
cular component. 

If  /?  is  a  whole  multiple  of  co,  or  even  nearly  so,  oscil- 
latory surging  of  the  current  of  corresponding  frequency 
will  occur  if  the  e.m.f.  has  a  harmonic  of  that  frequency. 


CHAPTER  VII 


COMPOUND  HARMONIC  CURRENT,  E.M.F.  AND 
POWER 


§  50.  Periodic  currents  (or  electromotive  forces)  which 
do  not  follow  simple  harmonic  laws  may  be  represented  by 
means  of  revolving  vectors.  The  most  evident  way  of 
representing  such  a  current  is  by  the  projection  of  a  line 
whose  length  equals  the  maximum  value  of  the  current, 
the  sign  not  being  considered.  The  angular  velocity  of 
rotation  may  be  varied  as  required,  at  times  if  necessary 
being  reduced  to  zero  or  even  reversed  in  sense.  Such 
a  method  while  involving  no  real  difficulty,  when  viewed 
from  a  purely  mechanical  standpoint,  in  fact  involves 
considerable  difficulty  when  an  attempt  is  made  to  express 
it  in  mathematical  symbols. 


USE   OF   FOURIER  SERIES 

§  51.  A  better  method  is  to  resolve  the  periodic  cur- 
rent into  a  number  of  harmonic  terms.  The  periodic 
current,  in  other  words,  may  be  expressed  as  a  Fourier 
series  of  the  form, 

l  cos  (a)t-di)  +  V2/2  cos  (2a)t-d2) 

+  V2/3  cos  (3a>t- d3)  +etc.+V2In  cos  (na>t-dn)  -J-etc. 

68 


USE  OF  FOURIER  SERIES  69 

If  preferred,  the  series  may  take  the  form 
sin  cot+A2  sin 


sn  ncut+etc., 

cos  wt+B2  cos  2a>t+B3  cos  3^+etc. 

+  Bn  sin  ncut+etc. 

The  relations  among  the  constants  of  the  two  equations 
are  evident.  The  latter  form  of  the  equation  has  for  our 
purposes  little  to  commend  it,  and  we  shall  not  use  it. 
The  former  form  expresses  the  current  as  a  constant  plus 
the  sum  of  the  projections  of  lines  of  length  V2/i,  \/2/2, 
V2/3,  etc.,  V2In,  etc.,  where  /i,  72,  /3,  etc.,  In,  etc.,  are 
the  effective  values  of  the  components  of  the  current. 

In  general,  currents  with  which  we  shall  deal  may  be 
represented  with  sufficient  approximation  by  a  very  limited 
number  of  terms;  and  in  most  cases  only  the  terms  of 
odd  order  are  present  in  currents  of  commercial  circuits. 
The  Fourier  series  in  such  cases  reduces  to 


cos  (wt-di)  +/3cos 

cos  (5cot-d5)  +etc.]. 


The  constant  term  70  is  only  present  in  case  the  average 
value  of  the  current  differs  from  zero.  The  terms  of  even 
order  are  present  if  successive  half  waves  differ  in  anything 
but  sign,  proper  allowance  being  made  for  the  constant 
term  if  present. 

It  is  evident  that  the  resultant  revolving  vector,  in 
cases  of  this  kind,  is  represented  by  a  broken  line,  each 
part  of  which  revolves  with  its  own  proper  angular  velocity. 

Similar  methods  may  be  used  to  express  a  periodic 
e.m.f.  Using  the  first  form  we  may  write 


i   COS  (a)t—  AI)  +  E2  COS  \£,uj(,—  /\2) 

cos  (3ajt—  ^3)  +etc.  +En  cos  (nwt  —  An)  +etc.]. 


70  REVOLVING  VECTORS 

It  is  assumed  that  current  and  e.m.f.  have  equal 
frequencies. 

§  52.  The  power  developed  in  the  circuit  is  found  by 
taking  the  product  of  e  and  i.  We  have  in  the  product 
a  number  of  terms  of  the  form, 

2EqIr  cos  (ra)t-dr)  cos  (qa>t-Xq). 
If  we  substitute  for  these  cosine  products  their  equal, 
EJr[cos  ( 


and  remember  that  the  sum  of  a  number  of  simple  har- 
monic quantities  of  the  same  frequency  is  another  simple 
harmonic  quantity  of  that  same  frequency,  the  expression 
for  p  reduces  to 


n  cos     n-   n  +i  cos       -i 

+  P2  cos  (2a)t-p2)  +PB  cos 
-fete.  +Pn  cos  (na>t  —  /3n)  +etc. 

The  average  power  is  made  up  of  the  constant  terms, 
for  the  variable  terms  (including  all  terms  functions  of  co) 
have  an  average  value  zero.  The  average  power  P  is 


cos    <i  -i 
-k2)  +etc.+EnIncos(dn-Xn)  +etc. 

If,  as  is  generally  true  of  commercial  circuits,  the  e.m.f. 
and  current  are  approximately  simple  harmonic  and  if 
the  only  harmonics  present  are  of  odd  order,  the  expression 
for  average  power  will  reduce  to  three  or  four  terms  whose 
sum  is  constant.  The  complete  expression,  practically  con- 
sidered, will  reduce  for  the  instantaneous  power  p  to 


cos     a)-2  +4  cos         -/4 

+P6cos 


POWER  FACTOR  71 

This  may  be  expressed  by  an  eccentric  revolving  vector 
whose  origin  is  distant  from  the  center  of  rotation  by  an 
amount  P 

POWER  FACTOR 

§  53.  For  the  power  factor  of  such  a  circuit  to  be  unity, 
it  is  necessary  that  the  current  and  e.m.f.  curves  be  pre- 
cisely similar,  i.e.,  the  current  at  every  instant  must  be  in 
the  same  fixed  proportion  to  the  e.m.f,  Mathematically 
expressed,  this  means  that 

E0:I0::  EI-.I!  ::E2:I2  ::E3:I3::  etc.  ::En:In  ::  etc., 

and  that  AI  =  £I,  ^2  =  ^2,  ^3=^3,  etc.,  ln  =  dn,  etc.  Under 
all  other  circumstances  the  product  of  the  effective  values 
of  current  and  e.m.f.  (/  and  E)  will  exceed  the  average 
power.  This  may  be  shown  as  follows : 

The  effective  value  /  of  the  current,  square  root  of  mean 
square  of  i,  is 


for  as  before  in  the  expression  for  average  power,  the  cross 
products  involving  different  frequencies  add  nothing  to  the 
final  result. 

In  the  same  way,  the  effective  value  E  of  the  e.m.f.  is 


E=  \/Eo2+Ei2+E22+E32+etc.  =  \2En2. 

o 

Let  us  suppose  the  various  components  of  the  current 
to  be  in  the  proportion 

/o : /i : I2 : /3 : etc.  : :  1  :a : /? :  f :  etc., 


72  REVOLVING  VECTORS 

and  those  of  the  e.m.f. 

E0:Ei:E2:E3:eic.  ::  1: 0:1:^1:7-1: etc., 

and  that  all  the  individual  power  factors  are  unity,  i.e., 
AI  =  $I,  ^2  =  ^2,  AS  =  £3,  etc.     We  then  have 

7  =  /0\/l+a2-h^2+7'2+etc., 


from  which  it  follows  that 


+/?iV+etc.] 


Canceling  like  plus  and  minus  quantities  and  combin- 
ing the  rest,  we  have 


fi2T2_p2 


As  the  right  side  of  the  equation  is  the  sum  of  squares, 
it  cannot  be  negative,  and  can  be  zero  only  if  a=«i,  /?=/?i, 
r  =  ri>  e^c-  Under  all  other  circumstances  we  must  have 
El  greater  than  P.  If  the  individual  power  factors  are  less 
than  unity,  the  value  of  P  will  be  smaller  still.  We  there- 
fore see  that  for  the  power  factor  of  the  effective  current 
to  be  unity,  the  current  and  e.m.f.  must  have  precisely 
similar  form. 


POWER  FACTOR  73 

§  54.  It  is  well  known  that  hysteresis  modifies  the  form 
of  the  current  curve,  so  that  it  cannot  be  as  a  rule  of  the 
same  form  as  the  e.m.f.  curve.  It  is  for  this  reason  usually 
impossible  precisely  to  obtain  unity  power  factor  in  the  case 
of  a  synchronous  motor  whose  field  has  been  adjusted  for 
maximum  power  factor;  for  no  one  field  excitation  can 
bring  all  instantaneous  values  of  current  and  e.m.f.  into  a 
constant  ratio. 


CHAPTER   VIII 
INTERLINKED  CIRCUITS,  MUTUAL   INDUCTION 

§  55.  When  two  circuits  are  linked  together  by  means 
of  the  magnetic  lines  of  force,  due  to  currents  in  both  or 
either  circuit,  we  may  observe  the  phenomenon  of  elec- 
tromotive forces  due  to  mutual  inductance.  This  phenom- 
enon was  first  observed  by  Joseph  Henry.  The  phenomenon 
of  self  inductance  was  discovered  by  Michael  Faraday,  who 
gave  to  the  world  the  invaluable  conception  of  lines 
of  force  to  explain  inductive  phenomena,  whether  electric 
or  magnetic  in  origin.  Faraday  was  not  a  mathematical 
physicist  and  his  famous  researches  with  respect  to  lines 
of  force  were  not  appreciated  until  put  into  mathematical 
language  by  Maxwell.  Unfortunately  for  our  present 
electromagnetic  terminology,  Maxwell  saw  fit  to  use  the 
expression  lines  of  induction  in  place  of  Faraday's  lines  of 
force  which  we  commonly  represent  by  0  (for  the  total 
flux)  and  B  (for  flux  per  unit  area).  Maxwell  also  un- 
fortunately gave  the  expression  lines  of  force  a  new  meaning, 
denoting  by  it  field  strength,  for  which  we  use  the  symbol 
H  (for  unit  area).  Many  writers  by  error  use  H  to  repre- 
sent Faraday's  lines  of  force  in  electromotive  force  for- 
mulae. It  is  evident  that  change  of  flux  (B),  not  field 
strength  (H),  determines  the  inductive  electromotive  force. 
Such  writers  use  0  to  represent  the  surface  integrals  of 
H  and  B  indifferently.  The  writer  in  company  with  many 
others  favors  holding  to  Faraday's  expression  lines  of  force 
to  represent  flux  (not  field  strength). 

74 


OHM'S  LAW  EXTENDED  TO  INDUCTIVE  CIRCUITS  75 


OHM'S    LAW    EXTENDED    TO  MUTUALLY    INDUCTIVE 

CIRCUITS 

§  56.  When  two  electric  circuits  are  interlinked  by  lines 
of  force,  we  have  a  new  term  in  the  law  of  Ohm  (extended 
to  variable  conditions).  The  expression  for  the  electro- 
motive force  ei  impressed  on  the  primary  circuit,  designated 
by  the  subscript  1,  becomes 


In  this  expression  #1  and  LI  are  the  resistance  and  self 
inductance  of  the  primary  coil,  i\  is  the  current  in  the 
primary  circuit,  M  is  the  mutual  inductance  between  the 
two  circuits  and  i2  is  the  current  in  the  secondary  circuit. 
The  e.m.f.  produced  in  the  secondary  circuit,  because  of 

rate  of  change  in  the  primary  circuit,  is  equal  to  —M—^. 

If  the  secondary  circuit  is  closed  and  a  current  i2  is  produced, 
a  part  of  this  e.m.f.  is  lost  in  ohmic  drop  of  potential 

i2R2  and  electromotive  force  of  self  induction  L2-^,  leaving 

ClfL 

available  at  the  terminals  the  remainder  e2,  impressed  by 
the  secondary  on  its  external  circuit.  We  have  then 

dii          .          di 


The  difference  in  the  form  of  the  two  equations  is  due 
to  the  conventional  agreement  to  consider  the  primary  e.m.f. 
to  be  applied  to  the  primary,  and  the  second  e.m.f.  to 
be  applied  by  the  secondary. 

§  57.  The  primary  and  secondary  e.m.f.'s  may  be 
expressed  in  terms  of  Faraday's  lines  of  force  also.  Calling 
the  flux  linking  with  the  primary  <£1;  that  linking  with  the 


76  REVOLVING  VECTORS 

secondary  <£2,  and  that  linking  with  both,  0,  with  maxi- 
mum values  of  4>i,  <P2,  and  0  respectively,  and  designating 
by  NI  and  N%  the  turns  of  wire  in  the  two  coils,  we  shall 
Jiave 


If  the  primary  and  secondary  circuits  were  so  closely 
related  that  (/>i=(f>2  =  (f>,  all  lines  of  flux  must  link  with 
every  turn  of  both  coils.  Wliile  in  fact  this  is  an  impos- 
sible assumption,  we  do  find  it  closely  approached  in  good 
commercial  transformers  with  moderate  loads. 

§  58.  The  method  of  revolving  vectors  may  be  used 
to  represent  the  electromotive  forces,  currents  and  fluxes. 
The  application  is  not  difficult  if  they  follow  harmonic 
laws.  If  all  are  simple  harmonic,  we  are  dealing  with  an 
ideal  transformer.  In  the  case  of  practical  transformers, 
we  find  that  even  if  the  electromotive  forces  are  simple 
harmonic,  the  currents  and  fluxes  will  follow  more  com- 
plicated laws. 

FARADAY'S   RING 

§  59.  Let  us  consider  the  simplest  form  of  transformer, 
a  Faraday  ring.  Suppose  the  core  built  up  of  thin  sheets 
of  extremely  soft  iron  of  high  permeability  /*.  We  shall 
assume  this  permeability  to  be  constant  for  all  values  of 
</>.  We  shall  also  assume  that  no  flux  leaves  the  ring, 
and  that  the  primary  and  secondary  windings  have  extremely 
low  resistance,  and  are  so  close  to  the  ring  and  so  well 
intermingled  that  no  flux  exists  outside  the  ring.  We 
shall  then  have  an  ideal  transformer.  Let  the  core  have 
a  permeability  /*,  an  average  length  K  and  a  cross-section 


FARADAY'S  RING  77 

A.  As  before  there  are  N\  primary  and  N2  secondary 
turns,  and  the  primary  and  secondary  currents  are  i\  and 
i2.  Let  us  first  suppose  i2  is  zero,  the  condition  of  an 
open  secondary  circuit.  The  magnetic  field  has  been 
supposed  to  satisfy  the  solenoidal  condition,  that  all  lines 
of  field  intensity  keep  within  the  core,  like  water  flowing 

r^ 

in  a  pipe.1     The  magnetic  reluctance  of  the  core  is  —r—  . 

Afj. 

The  magnetomotive  force  is 


and  it  produces  in  the  core  a  flux  of  lines  of  force  (Faraday's) 

.     4xNiAu.ii 

*-  -IT-"' 

with  a  flux  intensity  per  unit  cross-section  of  the  core, 

4xNi  fJL  li 

ir~- 

It  has  been  assumed  that  the  current  is  in  C.G.S.  units. 
If  ii  is  expressed  in  amperes,  we  must  write  IOK  in  the 
place  of  K  in  both  equations. 

The  inductive  electromotive  force  due  to  rate  of  change 

of  flux  is  as  before  —  NI-JJ-  in  the  primary,  and   —  N2-^- 

CIL  (It 

in  the  secondary.     We  therefore  have 


n  -       *T  D  •    ,    xiu.i   ,  , 

=  Riii  +  2vi~  =  R\i\  H  ---  ^—  -  -r-  (open  secondary)  , 

(Mi  J\.  (tt 


and 


.  , 

—  (open  secondary)  . 


1  The  words  solenoid  and  solenoidal  are  derived  from  the  Greek 
word  for  pipe  or  channel,  and  are  intended  to  convey  the  idea 
of  a  flux  keeping  within  its  channel. 


78  REVOLVING  VECTORS 

If  the  secondary   circuit  is   closed,  $  will  depend  on 
both  ii  and  i%  and  we  shall  have 


_  -    .         di\     ,,  dis 

and 


The  conclusions  reached  are  valid  only  when  //  is  con- 
stant and  the  flux  solenoidal,  i.e.,  there  is  no  leakage  of 
flux. 


CONCERNING  LINES   OF   FORCE 

§  60.  To  prove  that  it  is  flux  (Faraday's  lines  of  force), 
not  field  strength  (Maxwell's  lines  of  force),  which  deter- 
mines electromotive  force,  we  may  consider  two  trans- 
formers built  with  precisely  similar  dimensions,  their  only 
difference  being  that  one  has  a  well-laminated  soft  iron 
core  having  a  permeability  say  3000,  the  other  having  a 
wooden  core  of  permeability  1.  Let  the  secondaries  be 
connected  to  high-resistance  voltmeters,  but'  otherwise  have 
no  load.  The  secondary  currents  are  negligible.  Connect 
the  primaries  of  the  two  transformers  in  series  and  apply 
an  e.m.f.,  which  will  cause  full  secondary  voltage  in  the 
former.  The  secondary  voltage  of  the  latter  will  be  -g-gVir 
of  that  of  the  former.  The  currents  in  the  two  primaries 
are  alike  and  they  produce  equal  field  strength  H  in  both 
cores.  If  it  were  rate  of  field  strength  change  which 
determines  e.m.f.,  the  transformers  would  have  equal 
secondary  voltages.  We  find  on  the  contrary  that  the 
secondary  voltages  are  in  the  ratio  of  the  number  of 
Faraday's  lines  of  force.  From  this  we  conclude  that  it 
is  B,  not  H,  which  determines  electromotive  force. 


RATIO  OF  TRANSFORMATION  79 

RATIO   OF  TRANSFORMATION 

61.  We  showed  in  §  57  that  in  a  transformer 


and 


For  small  currents,  R\i\  and  R2i2  will  be  small.  If 
they  may  be  neglected,  and  if  there  is  negligible  loss  of 
flux,  so  that  we  may  consider  fa  and  <£2  equal,  we  obtain 
the  approximate  ratio 

ei  :e2::Ni  :-N2. 
If  ei  follows  a  simple  harmonic  law,  we  have 

(j)  =  @  cos  (o)t  —  _  )  =  $  sin  cut, 

and  neglecting  the  ohmic  drop  Riii  and  R2i2}  we  shall  have 
cos  wt  =  Nia>  0  cos  a>t 


and 

e2  =  \/2E2  cos  (cot—Ti)  =  —  N2(o0  cos  a)t; 

and  finally  for  the  ideal  ratio  of  transformation  of  elec- 
tromotive forces  (effective  values)  ,  we  shall  have 

E1:E2::N1:N2. 

In  practical  transformers  there  is  always  -some  loss  of 
flux  due  to  magnetic  leakage,  and  Riii  and  R2i2  cannot 
be  neglected.  For  these  reasons  the  secondary  electro- 
motive force  generally  falls  below  its  ideal  value. 


80  REVOLVING  VECTORS 

§  62.  In  our  ideal  transformer  we  have  assumed  the 
core  well  laminated  and  of  very  high  permeability.  If 
the  secondary  circuit  is  open  the  primary  will  require  an 
extremely  small  current  to  produce  the  necessary  flux. 
Had  the  core  not  been  laminated,  eddy  currents  generated 
in  the  core  would  have  required  more  primary  current. 
Also  had  the  permeability  of  the  core  been  low,  a  greater 
magnetizing  current  would  have  been  required. 

If  the  secondary  is  delivering  a  normal  current,  the 
magnetizing  effect  of  the  primary  and  secondary  currents 
will  practically  offset  one  another;  for  it  has  been  assumed 
that  owing  to  the  high  permeability  of  the  core,  very  little 
magnetizing  current  is  required.  The  magnetizing  effect 
is  proportional  to  the  aggregate  ampere-turns.  We  there- 
fore have  iiNi  practically  equal  to  —i2N2  at  all  times, 
or  for  effective  values,  neglecting  magnetizing  current, 

h:I2::N2:Nlt 


TRANSFORMER  DIAGRAMS,   LAGGING   CURRENT 

§  63.  The  phase  relation  of  current  and  e.m.f.,  primary 
and  secondary,  depends  on  the  exterior  portion  of  the 
secondary  circuit.  If  this  portion  of  the  circuit  is  a  non- 
inductive  resistance,  the  secondary  current  and  e.m.f.  will 
be  in  the  same  phase.  The  same  is  true  also  for  any 
arrangement  giving  unit  power  factor.  If  the  secondary 
current  is  out  of  phase  with  its  e.m.f.  we  may  have  either 
lead  or  lag. 

In  any  case  a  revolving  vector  diagram  may  be  used 
to  represent  the  facts.  To  illustrate  a  case  in  which  the 
current  lags  behind  the  e.m.f.,  let  us  draw  a  line  OC2  to 
represent  the  secondary  ampere-turns  I2N2  and  a  line 

7/1 

OB2  to  represent  the  secondary  volt s-per-t urn  -^  in  their 


TRANSFORMER  DIAGRAMS,   LAGGING  CURRENT    81 


proper   phase   relation    (Fig.    20).     Draw   parallel   to    OC2 
a  line  B2A2  to  represent  ohmic  drop  per  secondary  turn 

-p    r 

-£—  •     Then  OA2  will  represent  the  e.m.f.  per  secondary 

term  produced  by  the  varying  flux.     At  right   angles  to 
OA2  (90°  in  advance)  draw  OF  to  repre- 


sent the  flux  0.  Opposite  to  OA2, 
equal  to  it,  draw  OA\  to  represent  the 
part  of  the  applied  e.m.f.  required  per 
turn  to  balance  the  induced  e.m.f.  Draw 
the  line  OC\  equal  and  opposite  to  OC2 
to  represent  the  primary  ampere-turns 
IiNij  which  were  assumed  to  equal  the 
secondary  ampere-turns.  Draw  the  line 
A  iBi  parallel  to  OC\  to  represent  the 

7?  J 

primary  ohmic  drop  per  turn  -TT—  ,  and 

last  draw  the  line  OBi,  which  represents 
the  volts-per-turn  which  must  be  applied 
to  the  primary  by  some  external  source. 
Similar  quantities  must  be  drawn  to 
the  same  scale,  but  the  ampere-turns 
need  not  be  drawn  to  the  same  scale 
as  the  volts-per-turn.  The  reason  for  representing  ampere- 
turns  rather  than  amperes,  and  volts-per-turn  instead  of 
volts,  is  to  keep  the  lines  of  reasonable  length.  Other- 
wise in  a  transformer  with  high  ratio  of  transformation, 
similar  quantities  would  be  represented  by  lines  of  very 
inconvenient  lengths. 


82 


REVOLVING  VECTORS 


EXCITING    CURRENT,    CORE    LOSSES 

§  64.  If  the   core  losses   due  to    hysteresis   and    eddy 
currents  are  not  negligible,  the  primary  current  must  be 

increased  to  provide  for  these  losses  . 
This  additional  component  is  com- 
monly called  the  magnetizing  or 
exciting  current.  Strictly  speaking, 
the  magnetizing  current  is  only  the 
part  that  provides  for  hysteresis. 
The  magnetizing  current  is  never 
simple  harmonic,  as  hysteresis 
always  produces  a  certain  distor- 
tion in  the  wave  form.  We  may 
as  a  rule  ignore  the  higher  harmon- 
ics, as  they  are  practically  negligible 
in  comparison  with  the  load  cur- 
rents. This  may  be  illustrated  as 
follows:  If  1  1  represents  the  whole 
primary  current,  with  components 

AI  of  fundamental  frequency  and   A  3    of   three  times  as 
great  frequency,  and  if  higher  terms  are  negligible,  we  have 


FIG.  21. 


from  which  it  will  be  seen,  for  example,  that  if  A3  =  0.lAi, 
we  shall  have  /i  =  1.005  AI. 

To  provide  power  for  hysteresis  and  eddy  currents, 
the  exciting  current  must  be  ahead  of  the  flux  in  phase. 
In  Fig.  21  we  represent  the  ampere-turns  of  the  exciting 
current  by  the  line  Om.  The  line  OD  is  equal  and  opposite 
to  OC2;  and  OCi,  the  resultant  of  OD  and  Om,  represents 
the  total  primary  ampere-turns.  Put  in  another  way,  we 
may  say  that  Om  is  the  resultant  of  OCi  and 


EFFECT  OF  THE  FLUX  LEAKAGE 


83 


EFFECT  OF  THE  FLUX  LEAKAGE 

§  65.  In  case  some  of  the  lines  of  force  linking  with  the 
primary  coil  do  not  link  with  the  secondary  coil,  but 
pass  outside  the  core,  we  have  a  con- 
dition of  affairs  practically  equivalent 
to  considering  the  useful  flux  as  link- 
ing with  both  coils,  and  the  leakage 
flux  linking  with  a  choke  coil  in  series 
in  the  primary  circuit  and  having  a 
number  of  turns  equal  to  those  of 
the  primary.  The  choke  coil  is  sup- 
posed to  have  no  resistance.  The 
potential  drop  in  the  choke  coil  will 
lead  the  ohmic  drop  in  the  primary 
by  90°.  The  flux,  which  links  with 
a  portion  only  of  the  turns  in  either 
coil,  is  equivalent  to  a  smaller  amount 
linking  with  all  and  having  the  same 
total  amount  of  flux-turns. 

In  Fig.  22,  BiH,  which  leads  OCi 
(primary  ampere-turns)  by  90°,  repre- 
sents the  primary  volts-per-turn  con- 
sumed by  flux  leakage.  A\B\  is  the 

ohmic  drop  per  primary  turn.     OH  is  the  total  applied  e.m.f . 
per  primary  turn. 


iB, 


FIG.  22. 


TRANSFORMER   EQUATIONS 

§  66.  The  relations  among  the  quantities  represented 
by  Fig.  22  may  be  expressed  in  the  form  of  equations. 
Let  us  assume  that  the  external  portion  of  the  secondary 
circuit  has  an  impedance  r2+jx2;  and  let  us  assume  that 
the  flux  linking  with  the  primary  circuit,  but  not  with  the 


84 


REVOLVING  VECTORS 


secondary,  is  <£',  with  maximum  value  $'.  The  exciting 
current  is  m.  The  other  symbols  have  the  same  signif- 
icance as  before.  We  then  have  for  the  revolving  vectors, 


TRANSFORMER  DIAGRAMS,  LEADING  CURRENT 

§  67.  If  the  secondary  current  leads  its  electromotive 
force,   the   proper   phase   relations   are   to   be   taken   into 
consideration  in   making   the    dia- 
gram. 

Fig.  23  illustrates  the  case  of  a 
leading  secondary  current.  OC2 
represents  the  secondary  ampere- 
turns,  OB2  the  terminal  secondary 
volts-per-turn,  OC\  the  primary 
ampere-turns,  and  OH  the  primary 
terminal  volts-per-turn.  The  ohmic 
drop,  volts-per-turn,  is  represented 
by  ^2-^2,  in  the  secondary,  and 
by  A iBi,  in  the  primary.  The 
inductive  drop  in  the  primary, 
volts-per-turn,  is  represented  by 
BiH.  Because  of  the  phase  rela- 
tions, the  inductive  drop  in  Fig. 
23  is  not  a  drop  at  all,  but  rather 
a  rise;  for  the  terminal  volts-per- 
turn  OH  are  actually  less  than  the  amount  OBi}  which 
would  have  been  required  for  the  same  conditions  of  the 
secondary  with  no  magnetic  leakage. 


FIG.  23, 


DIFFICULTY  FOUND  IN  EXPONENTIAL  EXPRESSION   85 


DIFFICULTY    FOUND    IN    EXPONENTIAL  EXPRESSION 

§  68.  To  express  the  magnitudes  represented  in  the 
previous  equations  and  diagrams  in  terms  of  exponentials, 
the  phase  relations  must  all  be  determined.  If,  for  example, 
it  is  desired  to  express  the  secondary  current  in  terms  of 
the  primary  electromotive  force  and  the  constants  of  the 
circuit,  we  shall  find  the  exponential  expression  to  be  very 
complicated.  On  the  whole  it  is  better  in  a  numerical 
problem  to  proceed  through  the  series  of  equations  given 
in  §  66,  using  numerical  values.  Practically  it  is  difficult 
to  determine  the  leakage  flux  and  the  magnetizing  current. 
The  reason  is  because  the  permeability  of  the  core  is  not 
constant,  and  the  leakage  flux,  therefore,  is  higher  at  larger 
loads.  The  difficulties  are,  however,  not  insuperable. 


CONCLUSION 

§  69.  It  is  beyond  the  scope  of  this  small  book  to  con- 
sider all  the  alternating  current  problems  to  which  the 
method  of  revolving  vectors  may  be  applied.  If  the  reader 
has  become  well  enough  acquainted  with  the  method 
to  feel  confidence  in  applying  it  when  occasion  arises, 
the  author's  purpose  in  writing  on  the  subject  has  been 
accomplished. 


• 


INDEX 


PAGE 

Acceleration  in  spiral  motion 19 

in  uniform  circular  motion 18 

Aperiodic  discharge  of  a  condenser 54,  59 

Argand t 2 

Balanced  circuits 45  et  seq. 

Bernoulli 3 

Cauchy 2 

Complex  quantities iii 

"  "         addition  of iii,  23 

' '          division  of iii,  8 

1 '         multiplication  of iii,  8,  42 

powers  of 8 

roots  of 9 

rotary  power  of 5  et  seq. 

subtraction  of iii,  23 

Compound  harmonic  current,  e.m.f.  and  power 68 

Condenser,  aperiodic  or  non-oscillatory  discharge  of 54,  59 

oscillatory  discharge  of 52 

Core  losses 82 

Crank  diagram iv 

Current,  exciting  or  magnetizing 82 

' '         compound  harmonic 68 

' '         simple  harmonic 24 

' '         non-harmonic 52 

D'Alembert 3 

Damping,  effect  of 18 

De  Moivre 3 

Divided  circuit 32 

Discharge  of  condensers *.....     52 

Durege 3 

87 


88  INDEX 

PAGE 

Eccentric  uniform  circular  motion 42 

Electromotive  force,  compound  harmonic 68 

' '  simple  harmonic 22 

Euler 3,  12,  13,  21 

Exciting  current 82 

Exponential  expressions 11  et  seq. 

difficulties  found  in 85 

Fagnano 3 

Faraday 74,  78 

Faraday's  lines  of  force 78 

ring 74 

Flux-leakage,  effect  of 83 

Fourier  series,  use  of 68 

Four-phase  balanced  circuit 47 

Franpais 2 

Gauss 2 

Gergonne 2 

Harmonic  current,  compound 68 

"  simple 24 

e.m.f.,  compound 68 

"  "  simple ; ' 22 

Henry 74 

Hysteresis 73,  82 

Imaginary  exponents,  rotary  power  of 12 

Impedance 24 

Induction,  mutual 74 

self 25 

Interlinked  circuits 74 

Lagging  current 24,  80 

Leading  current 30,  84 

Lines  of  force,  Faraday's  and  Maxwell's 74 

' '  of  induction,  Maxwell's 74 

Magnetizing  current 82 

Maxwell 74 

Mutual  induction 74 

Negative  sign,  meaning  of 3 

Non-harmonic  current  and  e.m.f 52 

Non-oscillatory  discharge  of  condensers 54,  59 

Ohm's  law  extended  to  variable  current 25,  27,  61,  75 

Oscillatory  discharge  of  a  condenser 52 

Period  of  oscillatory  discharge 54 

Phenomena  observed  on  closing  the  circuit 60 


INDEX  89 

PAGE 

Polyphase  circuits,  balanced 45 

"  "        unbalanced 50 

Position  of  a  point  in  a  plane 14 

Power iii,  23,  40,  68 

Power  factor 71 

Powers  of  complex  quantities 8 

Ratio  of  transformation 79 

Resolution  into  components 35 

Rotary  power  of  complex  operators 5 

' '      of  imaginary  exponents 12 

"  "      of  roots  of  minus  one 1 

Royal  Academy  of  Science  and  Letters  of  Denmark 1 

Simple  harmonic  quantities 21 

Sign  of  perpendicularity 1 

Six-phase  balanced  circuit 48 

Solenoidal  condition 77 

Spiral  motion  and  quantities 18,  56,  65,  67 

Starting  term  on  closing  the  circuit 60 

Steinmetz iii,  iv,  23,  42 

Symmetrical  pair  of  triangles 38 

Three-phase  balanced  circuit 46 

Transformers,  diagrams,  and  equations 76  et  seq. 

Unbalanced  polyphase  circuits 50 

Uniform  circular  motion 15,  41 

Unit  power  factor 71 

Velocity  in  spiral  motion 18 

' '        in  uniform  circular  motion 16 

Wave  diagram iv 

Wessel iii,  1,  2. 


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